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Theorem List for Metamath Proof Explorer - 41501-41600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbcim2gVD 41501 Distribution of class substitution over a left-nested implication. Similar to sbcimg 3805. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcim2g 41164 is sbcim2gVD 41501 without virtual deductions and was automatically derived from sbcim2gVD 41501.
 1:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:: ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ) 3:1,2: ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))   ) 4:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ) 5:3,4: ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ) 6:5: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))   ) 7:: ⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ) 8:4,7: ⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))   ) 9:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)))   ) 10:8,9: ⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ) 11:10: ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)))   ) 12:6,11: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))   ) qed:12: ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))

TheoremsbcbiVD 41502 Implication form of sbcbii 3814. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcbi 41165 is sbcbiVD 41502 without virtual deductions and was automatically derived from sbcbiVD 41502.
 1:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:: ⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)    ▶   ∀𝑥(𝜑 ↔ 𝜓)   ) 3:1,2: ⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)    ▶   [𝐴 / 𝑥](𝜑 ↔ 𝜓)   ) 4:1,3: ⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)    ▶   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)   ) 5:4: ⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))   ) qed:5: ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

TheoremtrsbcVD 41503* Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trsbc 41166 is trsbcVD 41503 without virtual deductions and was automatically derived from trsbcVD 41503.
 1:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)   ) 3:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴)   ) 4:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝐴)   ) 5:1,2,3,4: ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))   ) 6:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)))   ) 7:5,6: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))   ) 8:: ⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) 9:7,8: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   ) 10:: ⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) 11:10: ⊢ ∀𝑥((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) 12:1,11: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   ) 13:9,12: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   ) 14:13: ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   ) 15:14: ⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   ) 16:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   ) 17:15,16: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   ) 18:17: ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑧([𝐴 / 𝑥]∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   ) 19:18: ⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑧[𝐴 / 𝑥]∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   ) 20:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑧∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   ) 21:19,20: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑧∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   ) 22:: ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) 23:21,22: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑧∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴)   ) 24:: ⊢ (Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) 25:24: ⊢ ∀𝑥(Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) 26:1,25: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]Tr 𝑥 ↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   ) 27:23,26: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴)   ) qed:27: ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))

TheoremtruniALTVD 41504* The union of a class of transitive sets is transitive. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. truniALT 41167 is truniALTVD 41504 without virtual deductions and was automatically derived from truniALTVD 41504.
 1:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴 Tr 𝑥   ) 2:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ) 3:2: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ 𝑦   ) 4:2: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑦 ∈ ∪ 𝐴   ) 5:4: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ) 6:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ) 7:6: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑦 ∈ 𝑞   ) 8:6: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑞 ∈ 𝐴   ) 9:1,8: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   ) 10:8,9: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   Tr 𝑞   ) 11:3,7,10: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ 𝑞   ) 12:11,8: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   ) 13:12: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 14:13: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 15:14: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 16:5,15: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   ) 17:16: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 18:17: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥    ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 19:18: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   Tr ∪ 𝐴   ) qed:19: ⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∪ 𝐴)
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

Theoremee33VD 41505 Non-virtual deduction form of e33 41360. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ee33 41147 is ee33VD 41505 without virtual deductions and was automatically derived from ee33VD 41505.
 h1:: ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) h2:: ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) h3:: ⊢ (𝜃 → (𝜏 → 𝜂)) 4:1,3: ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂)))) 5:4: ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) 6:2,5: ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) 7:6: ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) 8:7: ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) qed:8: ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜒𝜏)))    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))

TheoremtrintALTVD 41506* The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 41507. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trintALT 41507 is trintALTVD 41506 without virtual deductions and was automatically derived from trintALTVD 41506.
 1:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴Tr 𝑥   ) 2:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ) 3:2: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ 𝑦   ) 4:2: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑦 ∈ ∩ 𝐴   ) 5:4: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴𝑦 ∈ 𝑞   ) 6:5: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞)   ) 7:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑞 ∈ 𝐴   ) 8:7,6: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑦 ∈ 𝑞   ) 9:7,1: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   ) 10:7,9: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   Tr 𝑞   ) 11:10,3,8: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑧 ∈ 𝑞   ) 12:11: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   ) 13:12: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   ) 14:13: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴𝑧 ∈ 𝑞   ) 15:3,14: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ ∩ 𝐴   ) 16:15: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   ) 17:16: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   ) 18:17: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   Tr ∩ 𝐴   ) qed:18: ⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∩ 𝐴)
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

TheoremtrintALT 41507* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 41507 is an alternate proof of trint 5174. trintALT 41507 is trintALTVD 41506 without virtual deductions and was automatically derived from trintALTVD 41506 using the tools program translate..without..overwriting.cmd and the Metamath program "MM-PA> MINIMIZE_WITH *" command. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

Theoremundif3VD 41508 The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 4250. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. undif3 4250 is undif3VD 41508 without virtual deductions and was automatically derived from undif3VD 41508.
 1:: ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶))) 2:: ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) 3:2: ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 4:1,3: ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 5:: ⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   ) 6:5: ⊢ (   𝑥 ∈ 𝐴   ▶   (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)   ) 7:5: ⊢ (   𝑥 ∈ 𝐴   ▶   (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)   ) 8:6,7: ⊢ (   𝑥 ∈ 𝐴   ▶   ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))   ) 9:8: ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))) 10:: ⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ) 11:10: ⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   𝑥 ∈ 𝐵   ) 12:10: ⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   ¬ 𝑥 ∈ 𝐶    ) 13:11: ⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)   ) 14:12: ⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)   ) 15:13,14: ⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))   ) 16:15: ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))) 17:9,16: ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))) 18:: ⊢ (   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   ) 19:18: ⊢ (   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   𝑥 ∈ 𝐴   ) 20:18: ⊢ (   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   ¬ 𝑥 ∈ 𝐶    ) 21:18: ⊢ (   (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))   ) 22:21: ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 23:: ⊢ (   (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)   ▶   (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)   ) 24:23: ⊢ (   (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)   ▶   𝑥 ∈ 𝐴   ) 25:24: ⊢ (   (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)   ▶   (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))   ) 26:25: ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 27:10: ⊢ (   (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)   ▶   (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))   ) 28:27: ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 29:: ⊢ (   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)   ▶   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)   ) 30:29: ⊢ (   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)   ▶   𝑥 ∈ 𝐴   ) 31:30: ⊢ (   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)   ▶   (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))   ) 32:31: ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 33:22,26: ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 34:28,32: ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 35:33,34: ⊢ ((((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∨ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 36:: ⊢ ((((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∨ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))) 37:36,35: ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) 38:17,37: ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))) 39:: ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) 40:39: ⊢ (¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) 41:: ⊢ (¬ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴) ↔ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) 42:40,41: ⊢ (¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) 43:: ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 )) 44:43,42: ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴))) 45:: ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ ( 𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴))) 46:45,44: ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ ( (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))) 47:4,38: ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))) 48:46,47: ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))) 49:48: ⊢ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))) qed:49: ⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴))

TheoremsbcssgVD 41509 Virtual deduction proof of sbcssg 4446. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcssg 4446 is sbcssgVD 41509 without virtual deductions and was automatically derived from sbcssgVD 41509.
 1:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   ) 3:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)   ) 4:2,3: ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷 ))   ) 5:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷))   ) 6:4,5: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   ) 7:6: ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   ) 8:7: ⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷) )   ) 9:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))   ) 10:8,9: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷) )   ) 11:: ⊢ (𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) 110:11: ⊢ ∀𝑥(𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) 12:1,110: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))   ) 13:10,12: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   ) 14:: ⊢ (⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ ∀ 𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) 15:13,14: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷)   ) qed:15: ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋ 𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

TheoremcsbingVD 41510 Virtual deduction proof of csbin 4374. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbin 4374 is csbingVD 41510 without virtual deductions and was automatically derived from csbingVD 41510.
 1:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:: ⊢ (𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) } 20:2: ⊢ ∀𝑥(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} 30:1,20: ⊢ (   𝐴 ∈ 𝐵   ▶   [𝐴 / 𝑥](𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   ) 3:1,30: ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   ) 4:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   ) 5:3,4: ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   ) 6:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   ) 7:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)   ) 8:6,7: ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷 ))   ) 9:1: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷))   ) 10:9,8: ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   ) 11:10: ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   ) 12:11: ⊢ (   𝐴 ∈ 𝐵   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   ) 13:5,12: ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   ) 14:: ⊢ (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) = { 𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} 15:13,14: ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)   ) qed:15: ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ( ⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

TheoremonfrALTlem5VD 41511* Virtual deduction proof of onfrALTlem5 41168. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem5 41168 is onfrALTlem5VD 41511 without virtual deductions and was automatically derived from onfrALTlem5VD 41511.
 1:: ⊢ 𝑎 ∈ V 2:1: ⊢ (𝑎 ∩ 𝑥) ∈ V 3:2: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ (𝑎 ∩ 𝑥) = ∅) 4:3: ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ ¬ (𝑎 ∩ 𝑥) = ∅) 5:: ⊢ ((𝑎 ∩ 𝑥) ≠ ∅ ↔ ¬ (𝑎 ∩ 𝑥 ) = ∅) 6:4,5: ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ (𝑎 ∩ 𝑥) ≠ ∅) 7:2: ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]¬ 𝑏 = ∅) 8:: ⊢ (𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅) 9:8: ⊢ ∀𝑏(𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅) 10:2,9: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]¬ 𝑏 = ∅) 11:7,10: ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅) 12:6,11: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅ ↔ ( 𝑎 ∩ 𝑥) ≠ ∅) 13:2: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥 ) ↔ (𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥)) 14:12,13: ⊢ (([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅) ↔ ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅)) 15:2: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅)) 16:15,14: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) ↔ ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅)) 17:2: ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ( ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 ∩ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦) 18:2: ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 = (𝑎 ∩ 𝑥) 19:2: ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦 = 𝑦 20:18,19: ⊢ (⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 ∩ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦) = ((𝑎 ∩ 𝑥) ∩ 𝑦) 21:17,20: ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = (( 𝑎 ∩ 𝑥) ∩ 𝑦) 22:2: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅ ↔ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ⦋(𝑎 ∩ 𝑥) / 𝑏⦌ ∅) 23:2: ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌∅ = ∅ 24:21,23: ⊢ (⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ⦋(𝑎 ∩ 𝑥) / 𝑏⦌∅ ↔ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) 25:22,24: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅ ↔ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) 26:2: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ↔ 𝑦 ∈ (𝑎 ∩ 𝑥)) 27:25,26: ⊢ (([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ∧ [ (𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ (( 𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) 28:2: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ∧ [(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅)) 29:27,28: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) 30:29: ⊢ ∀𝑦([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) 31:30: ⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) 32:: ⊢ (∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ )) 33:31,32: ⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) 34:2: ⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ ( 𝑏 ∩ 𝑦) = ∅)) 35:33,34: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅) 36:: ⊢ (∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ ∃𝑦 (𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅)) 37:36: ⊢ ∀𝑏(∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅)) 38:2,37: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅)) 39:35,38: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) 40:16,39: ⊢ (([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) 41:2: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅)) qed:40,41: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥 )((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅) ↔ (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))

TheoremonfrALTlem4VD 41512* Virtual deduction proof of onfrALTlem4 41169. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem4 41169 is onfrALTlem4VD 41512 without virtual deductions and was automatically derived from onfrALTlem4VD 41512.
 1:: ⊢ 𝑦 ∈ V 2:1: ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋ 𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅) 3:1: ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌ 𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) 4:1: ⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎 5:1: ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦 6:4,5: ⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = ( 𝑎 ∩ 𝑦) 7:3,6: ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦) 8:1: ⊢ ⦋𝑦 / 𝑥⦌∅ = ∅ 9:7,8: ⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌ ∅ ↔ (𝑎 ∩ 𝑦) = ∅) 10:2,9: ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎 ∩ 𝑦) = ∅) 11:1: ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎) 12:11,10: ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥]( 𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) 13:1: ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅)) qed:13,12: ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))

TheoremonfrALTlem3VD 41513* Virtual deduction proof of onfrALTlem3 41170. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem3 41170 is onfrALTlem3VD 41513 without virtual deductions and was automatically derived from onfrALTlem3VD 41513.
 1:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ) 2:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ) 3:2: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   𝑥 ∈ 𝑎   ) 4:1: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ⊆ On   ) 5:3,4: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   𝑥 ∈ On   ) 6:5: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   Ord 𝑥   ) 7:6: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶    E We 𝑥   ) 8:: ⊢ (𝑎 ∩ 𝑥) ⊆ 𝑥 9:7,8: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶    E We (𝑎 ∩ 𝑥)   ) 10:9: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶    E Fr (𝑎 ∩ 𝑥)   ) 11:10: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∀𝑏((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅)   ) 12:: ⊢ 𝑥 ∈ V 13:12,8: ⊢ (𝑎 ∩ 𝑥) ∈ V 14:13,11: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   [(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅)   ) 15:: ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)( (𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) 16:14,15: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ ( 𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)   ) 17:: ⊢ (𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) 18:2: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ¬ (𝑎 ∩ 𝑥) = ∅   ) 19:18: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑎 ∩ 𝑥) ≠ ∅   ) 20:17,19: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅)   ) qed:16,20: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅   )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅   )

Theoremsimplbi2comtVD 41514 Virtual deduction proof of simplbi2comt 505. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. simplbi2comt 505 is simplbi2comtVD 41514 without virtual deductions and was automatically derived from simplbi2comtVD 41514.
 1:: ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜑 ↔ ( 𝜓 ∧ 𝜒))   ) 2:1: ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   ((𝜓 ∧ 𝜒 ) → 𝜑)   ) 3:2: ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜓 → (𝜒 → 𝜑))   ) 4:3: ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜒 → (𝜓 → 𝜑))   ) qed:4: ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑)))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))

TheoremonfrALTlem2VD 41515* Virtual deduction proof of onfrALTlem2 41172. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem2 41172 is onfrALTlem2VD 41515 without virtual deductions and was automatically derived from onfrALTlem2VD 41515.
 1:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ) 2:1: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ (𝑎 ∩ 𝑦)   ) 3:2: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ 𝑎   ) 4:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ) 5:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ) 6:5: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   𝑥 ∈ 𝑎   ) 7:4: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ⊆ On   ) 8:6,7: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   𝑥 ∈ On   ) 9:8: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   Ord 𝑥   ) 10:9: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   Tr 𝑥   ) 11:1: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑦 ∈ (𝑎 ∩ 𝑥)   ) 12:11: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑦 ∈ 𝑥   ) 13:2: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ 𝑦   ) 14:10,12,13: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ 𝑥   ) 15:3,14: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ (𝑎 ∩ 𝑥)   ) 16:13,15: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦))   ▶   𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)   ) 17:16: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦))   ) 18:17: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   ∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦))   ) 19:18: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦)   ) 20:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)   ) 21:20: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅   ) 22:19,21: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑎 ∩ 𝑦) = ∅   ) 23:20: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   𝑦 ∈ (𝑎 ∩ 𝑥)   ) 24:23: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   𝑦 ∈ 𝑎   ) 25:22,24: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅)   ▶   (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)   ) 26:25: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))   ) 27:26: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∀𝑦((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥 ) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))   ) 28:27: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   (∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥 ) ∩ 𝑦) = ∅) → ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))   ) 29:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦 ) = ∅   ) 30:29: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)   ) 31:28,30: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)   ) qed:31: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )

TheoremonfrALTlem1VD 41516* Virtual deduction proof of onfrALTlem1 41174. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem1 41174 is onfrALTlem1VD 41516 without virtual deductions and was automatically derived from onfrALTlem1VD 41516.
 1:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ) 2:1: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ) 3:2: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)    ) 4:: ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅ ) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) 5:4: ⊢ ∀𝑦([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) 6:5: ⊢ (∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) 7:3,6: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)   ) 8:: ⊢ (∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦( 𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) qed:7,8: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )

TheoremonfrALTVD 41517 Virtual deduction proof of onfrALT 41175. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALT 41175 is onfrALTVD 41517 without virtual deductions and was automatically derived from onfrALTVD 41517.
 1:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   ) 2:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   ) 3:1: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶    (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   ) 4:2: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶    ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   ) 5:: ⊢ ((𝑎 ∩ 𝑥) = ∅ ∨ ¬ (𝑎 ∩ 𝑥) = ∅) 6:5,4,3: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶    ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   ) 7:6: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   ) 8:7: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   ) 9:8: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   ) 10:: ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥𝑥 ∈ 𝑎) 11:9,10: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   ) 12:: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ) 13:12: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ≠ ∅   ) 14:13,11: ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   ) 15:14: ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) 16:15: ⊢ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) qed:16: ⊢ E Fr On
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
E Fr On

Theoremcsbeq2gVD 41518 Virtual deduction proof of csbeq2 3871. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbeq2 3871 is csbeq2gVD 41518 without virtual deductions and was automatically derived from csbeq2gVD 41518.
 1:: ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   ) 2:1: ⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥𝐵 = 𝐶 → [𝐴 / 𝑥] 𝐵 = 𝐶)   ) 3:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   ) 4:2,3: ⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥 ⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   ) qed:4: ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌ 𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

TheoremcsbsngVD 41519 Virtual deduction proof of csbsng 4629. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbsng 4629 is csbsngVD 41519 without virtual deductions and was automatically derived from csbsngVD 41519.
 1:: ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   ) 2:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   ) 3:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   ) 4:3: ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   ) 5:2,4: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   ) 6:5: ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   ) 7:6: ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   ) 8:1: ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   ) 9:7,8: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   ) 10:: ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵} 11:10: ⊢ ∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵} 12:1,11: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = ⦋ 𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   ) 13:9,12: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { 𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   ) 14:: ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} 15:13,14: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { ⦋𝐴 / 𝑥⦌𝐵}   ) qed:15: ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋ 𝐴 / 𝑥⦌𝐵})
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})

TheoremcsbxpgVD 41520 Virtual deduction proof of csbxp 5637. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbxp 5637 is csbxpgVD 41520 without virtual deductions and was automatically derived from csbxpgVD 41520.
 1:: ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   ) 2:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 3:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑤 = 𝑤   ) 4:3: ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 5:2,4: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 6:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   ) 7:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   ) 8:7: ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   ) 9:6,8: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   ) 10:5,9: ⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))   ) 11:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶))   ) 12:10,11: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))   ) 13:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑧 = ⟨𝑤   ,    𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩)   ) 14:12,13: ⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   ) 15:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   ) 16:14,15: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   ) 17:16: ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   ) 18:17: ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   ) 19:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   ) 20:18,19: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   ) 21:20: ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑤([𝐴 / 𝑥]∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   ) 22:21: ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   ) 23:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦 (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   ) 24:22,23: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   ) 25:24: ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑧([𝐴 / 𝑥]∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   ) 26:25: ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}    ) 27:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥] ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}   ) 28:26,27: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}    ) 29:: ⊢ {⟨𝑤   ,   𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} 30:: ⊢ (𝐵 × 𝐶) = {⟨𝑤   ,   𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} 31:29,30: ⊢ (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤 , 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} 32:31: ⊢ ∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} 33:1,32: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}   ) 34:28,33: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}   ) 35:: ⊢ {⟨𝑤   ,   𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} 36:: ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = { ⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)} 37:35,36: ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} 38:34,37: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶)   ) qed:38: ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))

TheoremcsbresgVD 41521 Virtual deduction proof of csbres 5843. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbres 5843 is csbresgVD 41521 without virtual deductions and was automatically derived from csbresgVD 41521.
 1:: ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   ) 2:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌V = V   ) 3:2: ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   ) 4:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)   ) 5:3,4: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   ) 6:5: ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   ) 7:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))   ) 8:6,7: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   ) 9:: ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) 10:9: ⊢ ∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) 11:1,10: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V))   ) 12:8,11: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   ) 13:: ⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) 14:12,13: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)   ) qed:14: ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

TheoremcsbrngVD 41522 Virtual deduction proof of csbrn 6047. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbrn 6047 is csbrngVD 41522 without virtual deductions and was automatically derived from csbrngVD 41522.
 1:: ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   ) 2:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤   ,   𝑦⟩ ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 3:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌⟨𝑤   ,   𝑦⟩ = ⟨𝑤, 𝑦⟩   ) 4:3: ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌⟨𝑤   ,   𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 5:2,4: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤   ,   𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 6:5: ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑤([𝐴 / 𝑥]⟨𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 7:6: ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 8:1: ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵)   ) 9:7,8: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤⟨𝑤    ,   𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 10:9: ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]∃𝑤 ⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 11:10: ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨ 𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   ) 12:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤 ⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   ) 13:11,12: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤 ⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   ) 14:: ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤   ,   𝑦⟩ ∈ 𝐵} 15:14: ⊢ ∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤   ,   𝑦⟩ ∈ 𝐵} 16:1,15: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   ) 17:13,16: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   ) 18:: ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤    ,   𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} 19:17,18: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋ 𝐴 / 𝑥⦌𝐵   ) qed:19: ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)

Theoremcsbima12gALTVD 41523 Virtual deduction proof of csbima12 5934. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbima12 5934 is csbima12gALTVD 41523 without virtual deductions and was automatically derived from csbima12gALTVD 41523.
 1:: ⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   ) 2:1: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ( ⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   ) 3:2: ⊢ (   𝐴 ∈ 𝐶   ▶    ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   ) 4:1: ⊢ (   𝐴 ∈ 𝐶   ▶    ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵)   ) 5:3,4: ⊢ (   𝐴 ∈ 𝐶   ▶    ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   ) 6:: ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵) 7:6: ⊢ ∀𝑥(𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵) 8:1,7: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋ 𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)   ) 9:5,8: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   ) 10:: ⊢ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) 11:9,10: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ( ⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)   ) qed:11: ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋ 𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

TheoremcsbunigVD 41524 Virtual deduction proof of csbuni 4853. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbuni 4853 is csbunigVD 41524 without virtual deductions and was automatically derived from csbunigVD 41524.
 1:: ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   ) 2:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)   ) 3:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)   ) 4:2,3: ⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   ) 5:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵))   ) 6:4,5: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   ) 7:6: ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   ) 8:7: ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   ) 9:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))   ) 10:8,9: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   ) 11:10: ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑧([𝐴 / 𝑥]∃𝑦( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))   ) 12:11: ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑧 ∣ [𝐴 / 𝑥]∃𝑦( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}   ) 13:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}    ) 14:12,13: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}   ) 15:: ⊢ ∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} 16:15: ⊢ ∀𝑥∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} 17:1,16: ⊢ (   𝐴 ∈ 𝑉   ▶   [𝐴 / 𝑥]∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}   ) 18:1,17: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌∪ 𝐵 = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}   ) 19:14,18: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}   ) 20:: ⊢ ∪ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)} 21:19,20: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵   ) qed:21: ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)

Theoremcsbfv12gALTVD 41525 Virtual deduction proof of csbfv12 6704. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbfv12 6704 is csbfv12gALTVD 41525 without virtual deductions and was automatically derived from csbfv12gALTVD 41525.
 1:: ⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   ) 2:1: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦} = { 𝑦}   ) 3:1: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵 }) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵})   ) 4:1: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { ⦋𝐴 / 𝑥⦌𝐵}   ) 5:4: ⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   ) 6:3,5: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵 }) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   ) 7:1: ⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦})   ) 8:6,2: ⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌(𝐹 “ { 𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   ) 9:7,8: ⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})    ) 10:9: ⊢ (   𝐴 ∈ 𝐶   ▶   ∀𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   ) 11:10: ⊢ (   𝐴 ∈ 𝐶   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   ) 12:1: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}   ) 13:11,12: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦 }}   ) 14:13: ⊢ (   𝐴 ∈ 𝐶   ▶   ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   ) 15:1: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   ) 16:14,15: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   ) 17:: ⊢ (𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} 18:17: ⊢ ∀𝑥(𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵 }) = {𝑦}} 19:1,18: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   ) 20:16,19: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   ) 21:: ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} 22:20,21: ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)   ) qed:22: ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

Theoremcon5VD 41526 Virtual deduction proof of con5 41148. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con5 41148 is con5VD 41526 without virtual deductions and was automatically derived from con5VD 41526.
 1:: ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   ) 2:1: ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜓 → 𝜑)   ) 3:2: ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → ¬ ¬ 𝜓 )   ) 4:: ⊢ (𝜓 ↔ ¬ ¬ 𝜓) 5:3,4: ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → 𝜓)   ) qed:5: ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))

TheoremrelopabVD 41527 Virtual deduction proof of relopab 5683. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. relopab 5683 is relopabVD 41527 without virtual deductions and was automatically derived from relopabVD 41527.
 1:: ⊢ (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   ) 2:1: ⊢ (   𝑦 = 𝑣   ▶   ⟨𝑥   ,   𝑦⟩ = ⟨𝑥   ,   𝑣 ⟩   ) 3:: ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   ) 4:3: ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥   ,   𝑣⟩ = ⟨ 𝑢, 𝑣⟩   ) 5:2,4: ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥   ,   𝑦⟩ = ⟨ 𝑢, 𝑣⟩   ) 6:5: ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥   ,   𝑦 ⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   ) 7:6: ⊢ (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,    𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   ) 8:7: ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦 ⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))) 9:8: ⊢ (∃𝑣𝑦 = 𝑣 → ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))) 90:: ⊢ (𝑣 = 𝑦 ↔ 𝑦 = 𝑣) 91:90: ⊢ (∃𝑣𝑣 = 𝑦 ↔ ∃𝑣𝑦 = 𝑣) 92:: ⊢ ∃𝑣𝑣 = 𝑦 10:91,92: ⊢ ∃𝑣𝑦 = 𝑣 11:9,10: ⊢ ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)) 12:11: ⊢ (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)) 13:: ⊢ (∃𝑣(𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢 , 𝑣⟩) → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)) 14:12,13: ⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)) 15:14: ⊢ (∃𝑢𝑥 = 𝑢 → ∃𝑢(𝑧 = ⟨𝑥   ,   𝑦 ⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)) 150:: ⊢ (𝑢 = 𝑥 ↔ 𝑥 = 𝑢) 151:150: ⊢ (∃𝑢𝑢 = 𝑥 ↔ ∃𝑢𝑥 = 𝑢) 152:: ⊢ ∃𝑢𝑢 = 𝑥 16:151,152: ⊢ ∃𝑢𝑥 = 𝑢 17:15,16: ⊢ ∃𝑢(𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑣𝑧 = ⟨ 𝑢, 𝑣⟩) 18:17: ⊢ (𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢∃𝑣𝑧 = ⟨ 𝑢, 𝑣⟩) 19:18: ⊢ (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑦∃𝑢 ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩) 20:: ⊢ (∃𝑦∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩) 21:19,20: ⊢ (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩) 22:21: ⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑥 ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩) 23:: ⊢ (∃𝑥∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩) 24:22,23: ⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢 ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩) 25:24: ⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩} 26:: ⊢ 𝑥 ∈ V 27:: ⊢ 𝑦 ∈ V 28:26,27: ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) 29:28: ⊢ (𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ (𝑧 = ⟨𝑥   ,   𝑦 ⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) 30:29: ⊢ (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) 31:30: ⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑥 ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) 32:31: ⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} = { 𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} 320:25,32: ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩} 33:: ⊢ 𝑢 ∈ V 34:: ⊢ 𝑣 ∈ V 35:33,34: ⊢ (𝑢 ∈ V ∧ 𝑣 ∈ V) 36:35: ⊢ (𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ (𝑧 = ⟨𝑢   ,   𝑣 ⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))) 37:36: ⊢ (∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))) 38:37: ⊢ (∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑢 ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))) 39:38: ⊢ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩} = { 𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))} 40:320,39: ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))} 41:: ⊢ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) } 42:: ⊢ {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) } 43:40,41,42: ⊢ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} 44:: ⊢ {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = (V × V) 45:43,44: ⊢ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} ⊆ (V × V) 46:28: ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V)) 47:46: ⊢ {⟨𝑥   ,   𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} 48:45,47: ⊢ {⟨𝑥   ,   𝑦⟩ ∣ 𝜑} ⊆ (V × V) qed:48: ⊢ Rel {⟨𝑥   ,   𝑦⟩ ∣ 𝜑}
(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Theorem19.41rgVD 41528 Virtual deduction proof of 19.41rg 41176. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. 19.41rg 41176 is 19.41rgVD 41528 without virtual deductions and was automatically derived from 19.41rgVD 41528. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) 2:1: ⊢ ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → ( 𝜑 ∧ 𝜓)))) 3:2: ⊢ ∀𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))) 4:3: ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑 ∧ 𝜓)))) 5:: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   ∀𝑥(𝜓 → ∀𝑥𝜓)   ) 6:4,5: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑 ∧ 𝜓)))   ) 7:: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ,   ∀𝑥𝜓   ▶    ∀𝑥𝜓   ) 8:6,7: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ,   ∀𝑥𝜓   ▶    ∀𝑥(𝜑 → (𝜑 ∧ 𝜓))   ) 9:8: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ,   ∀𝑥𝜓   ▶    (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))   ) 10:9: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))   ) 11:5: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀ 𝑥𝜓)   ) 12:10,11: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ( ∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))   ) 13:12: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))   ) 14:13: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥 𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))   ) qed:14: ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
(∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))

Theorem2pm13.193VD 41529 Virtual deduction proof of 2pm13.193 41178. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. 2pm13.193 41178 is 2pm13.193VD 41529 without virtual deductions and was automatically derived from 2pm13.193VD 41529. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ) 2:1: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ) 3:2: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   ) 4:1: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   ) 5:3,4: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   ) 6:5: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   ) 7:6: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   ) 8:2: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   ) 9:7,8: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)   ) 10:9: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝜑 ∧ 𝑦 = 𝑣)   ) 11:10: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝜑   ) 12:2,11: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ) 13:12: ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 14:: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ) 15:14: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ) 16:15: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   ) 17:14: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑    ) 18:16,17: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ( 𝜑 ∧ 𝑦 = 𝑣)   ) 19:18: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)   ) 20:15: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   ) 21:19: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑣 / 𝑦]𝜑   ) 22:20,21: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   ) 23:22: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   ) 24:23: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   ) 25:15,24: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ) 26:25: ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) qed:13,26: ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
(((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))

TheoremhbimpgVD 41530 Virtual deduction proof of hbimpg 41180. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbimpg 41180 is hbimpgVD 41530 without virtual deductions and was automatically derived from hbimpgVD 41530. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ) 2:1: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ∀𝑥(𝜑 → ∀𝑥𝜑)   ) 3:: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)), ¬ 𝜑   ▶   ¬ 𝜑   ) 4:2: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ∀𝑥(¬ 𝜑 → ∀𝑥¬ 𝜑)   ) 5:4: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   (¬ 𝜑 → ∀𝑥¬ 𝜑)   ) 6:3,5: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)), ¬ 𝜑   ▶   ∀𝑥¬ 𝜑   ) 7:: ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) 8:7: ⊢ (∀𝑥¬ 𝜑 → ∀𝑥(𝜑 → 𝜓)) 9:6,8: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)), ¬ 𝜑   ▶   ∀𝑥(𝜑 → 𝜓)   ) 10:9: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   (¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))   ) 11:: ⊢ (𝜓 → (𝜑 → 𝜓)) 12:11: ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓)) 13:1: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ∀𝑥(𝜓 → ∀𝑥𝜓)   ) 14:13: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   (𝜓 → ∀𝑥𝜓)   ) 15:14,12: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   (𝜓 → ∀𝑥(𝜑 → 𝜓))   ) 16:10,15: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ((¬ 𝜑 ∨ 𝜓) → ∀𝑥(𝜑 → 𝜓))   ) 17:: ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) 18:16,17: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))   ) 19:: ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥( 𝜑 → ∀𝑥𝜑)) 20:: ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ∀𝑥∀𝑥( 𝜓 → ∀𝑥𝜓)) 21:19,20: ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥(∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))) 22:21,18: ⊢ (   (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓))   ▶   ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))   ) qed:22: ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑𝜓) → ∀𝑥(𝜑𝜓)))

TheoremhbalgVD 41531 Virtual deduction proof of hbalg 41181. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbalg 41181 is hbalgVD 41531 without virtual deductions and was automatically derived from hbalgVD 41531. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(𝜑 → ∀𝑥𝜑)   ) 2:1: ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)   ) 3:: ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑) 4:2,3: ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)   ) 5:: ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦( 𝜑 → ∀𝑥𝜑)) 6:5,4: ⊢ (   ∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦(∀ 𝑦𝜑 → ∀𝑥∀𝑦𝜑)   ) qed:6: ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦 𝜑 → ∀𝑥∀𝑦𝜑))
(∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))

TheoremhbexgVD 41532 Virtual deduction proof of hbexg 41182. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbexg 41182 is hbexgVD 41532 without virtual deductions and was automatically derived from hbexgVD 41532. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(𝜑 → ∀𝑥𝜑)   ) 2:1: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 ∀𝑥(𝜑 → ∀𝑥𝜑)   ) 3:2: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (𝜑 → ∀𝑥𝜑)   ) 4:3: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (¬ 𝜑 → ∀𝑥¬ 𝜑)   ) 5:: ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦 ∀𝑥(𝜑 → ∀𝑥𝜑)) 6:: ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦 ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑)) 7:5: ⊢ (∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑)) 8:5,6,7: ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦 ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)) 9:8,4: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 ∀𝑥(¬ 𝜑 → ∀𝑥¬ 𝜑)   ) 10:9: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(¬ 𝜑 → ∀𝑥¬ 𝜑)   ) 11:10: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 (¬ 𝜑 → ∀𝑥¬ 𝜑)   ) 12:11: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 (∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑)   ) 13:12: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀ 𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑)   ) 14:: ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥 ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)) 15:13,14: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑)   ) 16:15: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 (¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   ) 17:16: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   ) 18:: ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦¬ 𝜑) 19:17,18: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   ) 20:18: ⊢ (∀𝑥∃𝑦𝜑 ↔ ∀𝑥¬ ∀𝑦¬ 𝜑) 21:19,20: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥∃𝑦𝜑)   ) 22:8,21: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑦 (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   ) 23:14,22: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   ) qed:23: ⊢ (   ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)   ▶   ∀𝑥 ∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)   )
(∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremax6e2eqVD 41533* The following User's Proof is a Virtual Deduction proof (see wvd1 41195) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2eq 41183 is ax6e2eqVD 41533 without virtual deductions and was automatically derived from ax6e2eqVD 41533. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥𝑥 = 𝑦   ) 2:: ⊢ (   ∀𝑥𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   ) 3:1: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   𝑥 = 𝑦   ) 4:2,3: ⊢ (   ∀𝑥𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   𝑦 = 𝑢   ) 5:2,4: ⊢ (   ∀𝑥𝑥 = 𝑦   ,   𝑥 = 𝑢   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 6:5: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))   ) 7:6: ⊢ (∀𝑥𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))) 8:7: ⊢ (∀𝑥∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑢))) 9:: ⊢ (∀𝑥𝑥 = 𝑦 ↔ ∀𝑥∀𝑥𝑥 = 𝑦) 10:8,9: ⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))) 11:1,10: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))   ) 12:11: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (∃𝑥𝑥 = 𝑢 → ∃𝑥 (𝑥 = 𝑢 ∧ 𝑦 = 𝑢))   ) 13:: ⊢ ∃𝑥𝑥 = 𝑢 14:13,12: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢 )   ) 140:14: ⊢ (∀𝑥𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ) 141:140: ⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)) 15:1,141: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 16:1,15: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 17:16: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 18:17: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 19:: ⊢ (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   ) 20:: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ) 21:20: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   𝑦 = 𝑢    ) 22:19,21: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   𝑦 = 𝑣    ) 23:20: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   𝑥 = 𝑢    ) 24:22,23: ⊢ (   𝑢 = 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ) 25:24: ⊢ (   𝑢 = 𝑣   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 26:25: ⊢ (   𝑢 = 𝑣   ▶   ∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 27:26: ⊢ (   𝑢 = 𝑣   ▶   (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 28:27: ⊢ (   𝑢 = 𝑣   ▶   ∀𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 29:28: ⊢ (   𝑢 = 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 30:29: ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢 ) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) 31:18,30: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (𝑢 = 𝑣 → ∃𝑥∃𝑦 (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) qed:31: ⊢ (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
(∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))

Theoremax6e2ndVD 41534* The following User's Proof is a Virtual Deduction proof (see wvd1 41195) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2nd 41184 is ax6e2ndVD 41534 without virtual deductions and was automatically derived from ax6e2ndVD 41534. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ ∃𝑦𝑦 = 𝑣 2:: ⊢ 𝑢 ∈ V 3:1,2: ⊢ (𝑢 ∈ V ∧ ∃𝑦𝑦 = 𝑣) 4:3: ⊢ ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) 5:: ⊢ (𝑢 ∈ V ↔ ∃𝑥𝑥 = 𝑢) 6:5: ⊢ ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) 7:6: ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦 (∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) 8:4,7: ⊢ ∃𝑦(∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) 9:: ⊢ (𝑧 = 𝑣 → ∀𝑥𝑧 = 𝑣) 10:: ⊢ (𝑦 = 𝑣 → ∀𝑧𝑦 = 𝑣) 11:: ⊢ (   𝑧 = 𝑦   ▶   𝑧 = 𝑦   ) 12:11: ⊢ (   𝑧 = 𝑦   ▶   (𝑧 = 𝑣 ↔ 𝑦 = 𝑣)   ) 120:11: ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣)) 13:9,10,120: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥𝑦 = 𝑣)) 14:: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ¬ ∀𝑥𝑥 = 𝑦   ) 15:14,13: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣)   ) 16:15: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥𝑦 = 𝑣)) 17:16: ⊢ (∀𝑥¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥𝑦 = 𝑣)) 18:: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦 ) 19:17,18: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀ 𝑥𝑦 = 𝑣)) 20:14,19: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥(𝑦 = 𝑣 → ∀𝑥𝑦 = 𝑣)   ) 21:20: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ((∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 22:21: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ((∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) 23:22: ⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) 24:: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦 ) 25:23,24: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) 26:14,25: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ∀𝑦((∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 27:26: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   (∃𝑦(∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))   ) 28:8,27: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ) 29:28: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ) qed:29: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
(¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))

Theoremax6e2ndeqVD 41535* The following User's Proof is a Virtual Deduction proof (see wvd1 41195) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. ax6e2eq 41183 is ax6e2ndeqVD 41535 without virtual deductions and was automatically derived from ax6e2ndeqVD 41535. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   𝑢 ≠ 𝑣   ▶   𝑢 ≠ 𝑣   ) 2:: ⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ) 3:2: ⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   𝑥 = 𝑢   ) 4:1,3: ⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   𝑥 ≠ 𝑣   ) 5:2: ⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   𝑦 = 𝑣   ) 6:4,5: ⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶   𝑥 ≠ 𝑦   ) 7:: ⊢ (∀𝑥𝑥 = 𝑦 → 𝑥 = 𝑦) 8:7: ⊢ (¬ 𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦) 9:: ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦) 10:8,9: ⊢ (𝑥 ≠ 𝑦 → ¬ ∀𝑥𝑥 = 𝑦) 11:6,10: ⊢ (   𝑢 ≠ 𝑣   ,   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ▶    ¬ ∀𝑥𝑥 = 𝑦   ) 12:11: ⊢ (   𝑢 ≠ 𝑣   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   ) 13:12: ⊢ (   𝑢 ≠ 𝑣   ▶   ∀𝑥((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   ) 14:13: ⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥¬ ∀𝑥𝑥 = 𝑦)   ) 15:: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦 ) 19:15: ⊢ (∃𝑥¬ ∀𝑥𝑥 = 𝑦 ↔ ¬ ∀𝑥𝑥 = 𝑦) 20:14,19: ⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   ) 21:20: ⊢ (   𝑢 ≠ 𝑣   ▶   ∀𝑦(∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   ) 22:21: ⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦)   ) 23:: ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ ∃ 𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) 24:22,23: ⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦)   ) 25:: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦 ) 26:25: ⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∃𝑦∀𝑦¬ ∀𝑥𝑥 = 𝑦) 260:: ⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦∀𝑦¬ ∀𝑥𝑥 = 𝑦) 27:260: ⊢ (∃𝑦∀𝑦¬ ∀𝑥𝑥 = 𝑦 ↔ ∀𝑦¬ ∀𝑥𝑥 = 𝑦) 270:26,27: ⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥 𝑥 = 𝑦) 28:: ⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦 ) 29:270,28: ⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦 ) 30:24,29: ⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦)   ) 31:30: ⊢ (   𝑢 ≠ 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣))   ) 32:31: ⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣))) 33:: ⊢ (   𝑢 = 𝑣   ▶   𝑢 = 𝑣   ) 34:33: ⊢ (   𝑢 = 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑢 = 𝑣)   ) 35:34: ⊢ (   𝑢 = 𝑣   ▶   (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣))   ) 36:35: ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣))) 37:: ⊢ (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣) 38:32,36,37: ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ( ¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)) 39:: ⊢ (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦 (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) 40:: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) 41:40: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃ 𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) 42:: ⊢ (∀𝑥𝑥 = 𝑦 ∨ ¬ ∀𝑥𝑥 = 𝑦) 43:39,41,42: ⊢ (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣 )) 44:40,43: ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ∃𝑥 ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) qed:38,44: ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥 ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))

Theorem2sb5ndVD 41536* The following User's Proof is a Virtual Deduction proof (see wvd1 41195) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. 2sb5nd 41186 is 2sb5ndVD 41536 without virtual deductions and was automatically derived from 2sb5ndVD 41536. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 2:1: ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 3:: ⊢ ([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑) 4:3: ⊢ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑) 5:4: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥] ∀𝑦[𝑣 / 𝑦]𝜑) 6:: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ¬ ∀𝑥𝑥 = 𝑦   ) 7:: ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑥𝑥 = 𝑦) 8:7: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑦𝑦 = 𝑥) 9:6,8: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ¬ ∀𝑦𝑦 = 𝑥   ) 10:9: ⊢ ([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀ 𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) 11:5,10: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) 12:11: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 13:: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) 14:: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥𝑥 = 𝑦   ) 15:14: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (∀𝑥[𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ) 16:13,15: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦 ]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ) 17:16: ⊢ (∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦] 𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 19:12,17: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) 20:19: ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 21:2,20: ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 22:21: ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 23:13: ⊢ (∃𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 24:22,23: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 240:24: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ( ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) 241:: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ( ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 242:241,240: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) 243:: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ( [𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) ↔ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))) 25:242,243: ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ([ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) 26:: ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥 ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) qed:25,26: ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))

Theorem2uasbanhVD 41537* The following User's Proof is a Virtual Deduction proof (see wvd1 41195) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. 2uasbanh 41187 is 2uasbanhVD 41537 without virtual deductions and was automatically derived from 2uasbanhVD 41537. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 h1:: ⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) 100:1: ⊢ (𝜒 → (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) 2:100: ⊢ (   𝜒   ▶   (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))   ) 3:2: ⊢ (   𝜒   ▶   ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ) 4:3: ⊢ (   𝜒   ▶   ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣 )   ) 5:4: ⊢ (   𝜒   ▶   (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)    ) 6:5: ⊢ (   𝜒   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))   ) 7:3,6: ⊢ (   𝜒   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   ) 8:2: ⊢ (   𝜒   ▶   ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)   ) 9:5: ⊢ (   𝜒   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜓 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))   ) 10:8,9: ⊢ (   𝜒   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜓   ) 101:: ⊢ ([𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓)) 102:101: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓)) 103:: ⊢ ([𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦 ]𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓)) 104:102,103: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓)) 11:7,10,104: ⊢ (   𝜒   ▶   [𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓)   ) 110:5: ⊢ (   𝜒   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)))   ) 12:11,110: ⊢ (   𝜒   ▶   ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓))   ) 120:12: ⊢ (𝜒 → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ) ∧ (𝜑 ∧ 𝜓))) 13:1,120: ⊢ ((∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓))) 14:: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓 ))   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓))   ) 15:14: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓 ))   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   ) 16:14: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓 ))   ▶   (𝜑 ∧ 𝜓)   ) 17:16: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓 ))   ▶   𝜑   ) 18:15,17: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓 ))   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ) 19:18: ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓 )) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 20:19: ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 21:20: ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ( 𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 22:16: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓 ))   ▶   𝜓   ) 23:15,22: ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓 ))   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)   ) 24:23: ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓 )) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) 25:24: ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) 26:25: ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ( 𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) 27:21,26: ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ( 𝜑 ∧ 𝜓)) → (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦( (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) qed:13,27: ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ( 𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦( (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
(𝜒 ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))       (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))

Theoreme2ebindVD 41538 The following User's Proof is a Virtual Deduction proof (see wvd1 41195) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. e2ebind 41189 is e2ebindVD 41538 without virtual deductions and was automatically derived from e2ebindVD 41538.
 1:: ⊢ (𝜑 ↔ 𝜑) 2:1: ⊢ (∀𝑦𝑦 = 𝑥 → (𝜑 ↔ 𝜑)) 3:2: ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑 )) 4:: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   ∀𝑦𝑦 = 𝑥   ) 5:3,4: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   (∃𝑦𝜑 ↔ ∃𝑥 𝜑)   ) 6:: ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑦∀𝑦𝑦 = 𝑥) 7:5,6: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   ∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑)   ) 8:7: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)   ) 9:: ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) 10:8,9: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑)   ) 11:: ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑) 12:11: ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑) 13:10,12: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)   ) 14:13: ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃ 𝑦𝜑)) 15:: ⊢ (∀𝑦𝑦 = 𝑥 ↔ ∀𝑥𝑥 = 𝑦) qed:14,15: ⊢ (∀𝑥𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃ 𝑦𝜑))
(Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))

20.36.8  Virtual Deduction transcriptions of textbook proofs

Theoremsb5ALTVD 41539* The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Unit 20 Excercise 3.a., which is sb5 2278, found in the "Answers to Starred Exercises" on page 457 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. The same proof may also be interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sb5ALT 41151 is sb5ALTVD 41539 without virtual deductions and was automatically derived from sb5ALTVD 41539.
 1:: ⊢ (   [𝑦 / 𝑥]𝜑   ▶   [𝑦 / 𝑥]𝜑   ) 2:: ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 3:1,2: ⊢ (   [𝑦 / 𝑥]𝜑   ▶   [𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑)   ) 4:3: ⊢ (   [𝑦 / 𝑥]𝜑   ▶   ∃𝑥(𝑥 = 𝑦 ∧ 𝜑 )   ) 5:4: ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ) 6:: ⊢ (   ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)   ▶   ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)   ) 7:: ⊢ (   ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)   ,   (𝑥 = 𝑦 ∧ 𝜑 )   ▶   (𝑥 = 𝑦 ∧ 𝜑)   ) 8:7: ⊢ (   ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)   ,   (𝑥 = 𝑦 ∧ 𝜑 )   ▶   𝜑   ) 9:7: ⊢ (   ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)   ,   (𝑥 = 𝑦 ∧ 𝜑 )   ▶   𝑥 = 𝑦   ) 10:8,9: ⊢ (   ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)   ,   (𝑥 = 𝑦 ∧ 𝜑 )   ▶   [𝑦 / 𝑥]𝜑   ) 101:: ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) 11:101,10: ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑 ) 12:5,11: ⊢ (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑 )) ∧ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)) qed:12: ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremvk15.4jVD 41540 The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Unit 15 Excercise 4.f. found in the "Answers to Starred Exercises" on page 442 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. The same proof may also be interpreted to be a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. vk15.4j 41154 is vk15.4jVD 41540 without virtual deductions and was automatically derived from vk15.4jVD 41540. Step numbers greater than 25 are additional steps necessary for the sequent calculus proof not contained in the Fitch-style proof. Otherwise, step i of the User's Proof corresponds to step i of the Fitch-style proof.
 h1:: ⊢ ¬ (∃𝑥¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) h2:: ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏 )) h3:: ⊢ ¬ ∀𝑥(𝜏 → 𝜑) 4:: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ∃𝑥¬ 𝜃   ) 5:4: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∀𝑥𝜃   ) 6:3: ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑) 7:: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜏 ∧ ¬ 𝜑)   ) 8:7: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   𝜏   ) 9:7: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ 𝜑   ) 10:5: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   𝜃   ) 11:10,8: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   (𝜃 ∧ 𝜏)   ) 12:11: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥(𝜃 ∧ 𝜏)   ) 13:12: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏)   ) 14:2,13: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ¬ ∀𝑥𝜒   ) 140:: ⊢ (∃𝑥¬ 𝜃 → ∀𝑥∃𝑥¬ 𝜃 ) 141:140: ⊢ (¬ ∃𝑥¬ 𝜃 → ∀𝑥¬ ∃𝑥 ¬ 𝜃) 142:: ⊢ (∀𝑥𝜒 → ∀𝑥∀𝑥𝜒) 143:142: ⊢ (¬ ∀𝑥𝜒 → ∀𝑥¬ ∀𝑥𝜒 ) 144:6,14,141,143: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ∀𝑥𝜒    ) 15:1: ⊢ (¬ ∃𝑥¬ 𝜑 ∨ ¬ ∃𝑥(𝜓 ∧ ¬ 𝜒)) 16:9: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   (𝜏 ∧ ¬ 𝜑)   ▶   ∃𝑥¬ 𝜑   ) 161:: ⊢ (∃𝑥¬ 𝜑 → ∀𝑥∃𝑥¬ 𝜑 ) 162:6,16,141,161: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∃𝑥¬ 𝜑    ) 17:162: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ¬ ∃𝑥 ¬ 𝜑   ) 18:15,17: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ∃𝑥( 𝜓 ∧ ¬ 𝜒)   ) 19:18: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∀𝑥(𝜓 → 𝜒)   ) 20:144: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∃𝑥¬ 𝜒    ) 21:: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶   ¬ 𝜒   ) 22:19: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   (𝜓 → 𝜒 )   ) 23:21,22: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶   ¬ 𝜓   ) 24:23: ⊢ (   ¬ ∃𝑥¬ 𝜃   ,   ¬ 𝜒   ▶   ∃ 𝑥¬ 𝜓   ) 240:: ⊢ (∃𝑥¬ 𝜓 → ∀𝑥∃𝑥¬ 𝜓 ) 241:20,24,141,240: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ∃𝑥¬ 𝜓    ) 25:241: ⊢ (   ¬ ∃𝑥¬ 𝜃   ▶   ¬ ∀𝑥𝜓    ) qed:25: ⊢ (¬ ∃𝑥¬ 𝜃 → ¬ ∀𝑥𝜓)
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))    &   (∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏))    &    ¬ ∀𝑥(𝜏𝜑)       (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

TheoremnotnotrALTVD 41541 The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Theorem 5 of Section 14 of [Margaris] p. 59 (which is notnotr 132). The same proof may also be interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. notnotrALT 41155 is notnotrALTVD 41541 without virtual deductions and was automatically derived from notnotrALTVD 41541. Step i of the User's Proof corresponds to step i of the Fitch-style proof.
 1:: ⊢ (   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜑   ) 2:: ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑)) 3:1: ⊢ (   ¬ ¬ 𝜑   ▶   (¬ 𝜑 → ¬ ¬ ¬ 𝜑)   ) 4:: ⊢ ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑 → 𝜑)) 5:3: ⊢ (   ¬ ¬ 𝜑   ▶   (¬ ¬ 𝜑 → 𝜑)   ) 6:5,1: ⊢ (   ¬ ¬ 𝜑   ▶   𝜑   ) qed:6: ⊢ (¬ ¬ 𝜑 → 𝜑)
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)

Theoremcon3ALTVD 41542 The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Theorem 7 of Section 14 of [Margaris] p. 60 (which is con3 156). The same proof may also be interpreted to be a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con3ALT2 41156 is con3ALTVD 41542 without virtual deductions and was automatically derived from con3ALTVD 41542. Step i of the User's Proof corresponds to step i of the Fitch-style proof.
 1:: ⊢ (   (𝜑 → 𝜓)   ▶   (𝜑 → 𝜓)   ) 2:: ⊢ (   (𝜑 → 𝜓)   ,   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜑   ) 3:: ⊢ (¬ ¬ 𝜑 → 𝜑) 4:2: ⊢ (   (𝜑 → 𝜓)   ,   ¬ ¬ 𝜑   ▶   𝜑   ) 5:1,4: ⊢ (   (𝜑 → 𝜓)   ,   ¬ ¬ 𝜑   ▶   𝜓   ) 6:: ⊢ (𝜓 → ¬ ¬ 𝜓) 7:6,5: ⊢ (   (𝜑 → 𝜓)   ,   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜓   ) 8:7: ⊢ (   (𝜑 → 𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓 )   ) 9:: ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑)) 10:8: ⊢ (   (𝜑 → 𝜓)   ▶   (¬ 𝜓 → ¬ 𝜑)   ) qed:10: ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

20.36.9  Theorems proved using conjunction-form Virtual Deduction

TheoremelpwgdedVD 41543 Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4525. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 41190 is elpwgdedVD 41543 using conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝐴 ∈ V   )    &   (   𝜓   ▶   𝐴𝐵   )       (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )

Theoremsspwimp 41544 If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. For the biconditional, see sspwb 5329. The proof sspwimp 41544, using conventional notation, was translated from virtual deduction form, sspwimpVD 41545, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

TheoremsspwimpVD 41545 The following User's Proof is a Virtual Deduction proof (see wvd1 41195) using conjunction-form virtual hypothesis collections. It was completed manually, but has the potential to be completed automatically by a tools program which would invoke Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimp 41544 is sspwimpVD 41545 without virtual deductions and was derived from sspwimpVD 41545. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   𝐴 ⊆ 𝐵   ▶   𝐴 ⊆ 𝐵   ) 2:: ⊢ (   .............. 𝑥 ∈ 𝒫 𝐴    ▶   𝑥 ∈ 𝒫 𝐴   ) 3:2: ⊢ (   .............. 𝑥 ∈ 𝒫 𝐴    ▶   𝑥 ⊆ 𝐴   ) 4:3,1: ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ⊆ 𝐵   ) 5:: ⊢ 𝑥 ∈ V 6:4,5: ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵    ) 7:6: ⊢ (   𝐴 ⊆ 𝐵   ▶   (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)    ) 8:7: ⊢ (   𝐴 ⊆ 𝐵   ▶   ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   ) 9:8: ⊢ (   𝐴 ⊆ 𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   ) qed:9: ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Theoremsspwimpcf 41546 If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimpcf 41546, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 41547, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

TheoremsspwimpcfVD 41547 The following User's Proof is a Virtual Deduction proof (see wvd1 41195) using conjunction-form virtual hypothesis collections. It was completed automatically by a tools program which would invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimpcf 41546 is sspwimpcfVD 41547 without virtual deductions and was derived from sspwimpcfVD 41547. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   𝐴 ⊆ 𝐵   ▶   𝐴 ⊆ 𝐵   ) 2:: ⊢ (   ........... 𝑥 ∈ 𝒫 𝐴    ▶   𝑥 ∈ 𝒫 𝐴   ) 3:2: ⊢ (   ........... 𝑥 ∈ 𝒫 𝐴    ▶   𝑥 ⊆ 𝐴   ) 4:3,1: ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ⊆ 𝐵   ) 5:: ⊢ 𝑥 ∈ V 6:4,5: ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵    ) 7:6: ⊢ (   𝐴 ⊆ 𝐵   ▶   (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)    ) 8:7: ⊢ (   𝐴 ⊆ 𝐵   ▶   ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   ) 9:8: ⊢ (   𝐴 ⊆ 𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   ) qed:9: ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

TheoremsuctrALTcf 41548 The sucessor of a transitive class is transitive. suctrALTcf 41548, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 41549, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)

TheoremsuctrALTcfVD 41549 The following User's Proof is a Virtual Deduction proof (see wvd1 41195) using conjunction-form virtual hypothesis collections. The conjunction-form version of completeusersproof.cmd. It allows the User to avoid superflous virtual hypotheses. This proof was completed automatically by a tools program which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. suctrALTcf 41548 is suctrALTcfVD 41549 without virtual deductions and was derived automatically from suctrALTcfVD 41549. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   Tr 𝐴   ▶   Tr 𝐴   ) 2:: ⊢ (   ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ) 3:2: ⊢ (   ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ 𝑦   ) 4:: ⊢ (   ................................... ....... 𝑦 ∈ 𝐴   ▶   𝑦 ∈ 𝐴   ) 5:1,3,4: ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) , 𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ 𝐴   ) 6:: ⊢ 𝐴 ⊆ suc 𝐴 7:5,6: ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) , 𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ suc 𝐴   ) 8:7: ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)    )   ▶   (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)   ) 9:: ⊢ (   ................................... ...... 𝑦 = 𝐴   ▶   𝑦 = 𝐴   ) 10:3,9: ⊢ (   ........ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧 ∈ 𝐴   ) 11:10,6: ⊢ (   ........ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   ) 12:11: ⊢ (   .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴)   ) 13:2: ⊢ (   .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   ) 14:13: ⊢ (   .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)   ) 15:8,12,14: ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)    )   ▶   𝑧 ∈ suc 𝐴   ) 16:15: ⊢ (   Tr 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   ) 17:16: ⊢ (   Tr 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   ) 18:17: ⊢ (   Tr 𝐴   ▶   Tr suc 𝐴   ) qed:18: ⊢ (Tr 𝐴 → Tr suc 𝐴)
(Tr 𝐴 → Tr suc 𝐴)

20.36.10  Theorems with a VD proof in conventional notation derived from a VD proof

TheoremsuctrALT3 41550 The successor of a transitive class is transitive. suctrALT3 41550 is the completed proof in conventional notation of the Virtual Deduction proof https://us.metamath.org/other/completeusersproof/suctralt3vd.html 41550. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 41195 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction (e.g., the sub-theorem whose assertion is step 19 used jaoded 41192). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem (e.g., the sub-theorem whose assertion is step 24 used dftr2 5160) . (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)

TheoremsspwimpALT 41551 If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimpALT 41551 is the completed proof in conventional notation of the Virtual Deduction proof https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html 41551. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 41195 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction (e.g., the sub-theorem whose assertion is step 9 used elpwgded 41190). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem (e.g., the sub-theorem whose assertion is step 5 used elpwi 4531). (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

TheoremunisnALT 41552 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. The User manually input on a mmj2 Proof Worksheet, without labels, all steps of unisnALT 41552 except 1, 11, 15, 21, and 30. With execution of the mmj2 unification command, mmj2 could find labels for all steps except for 2, 12, 16, 22, and 31 (and the then non-existing steps 1, 11, 15, 21, and 30). mmj2 could not find reference theorems for those five steps because the hypothesis field of each of these steps was empty and none of those steps unifies with a theorem in set.mm. Each of these five steps is a semantic variation of a theorem in set.mm and is 2-step provable. mmj2 does not have the ability to automatically generate the semantic variation in set.mm of a theorem in a mmj2 Proof Worksheet unless the theorem in the Proof Worksheet is labeled with a 1-hypothesis deduction whose hypothesis is a theorem in set.mm which unifies with the theorem in the Proof Worksheet. The stepprover.c program, which invokes mmj2, has this capability. stepprover.c automatically generated steps 1, 11, 15, 21, and 30, labeled all steps, and generated the RPN proof of unisnALT 41552. Roughly speaking, stepprover.c added to the Proof Worksheet a labeled duplicate step of each non-unifying theorem for each label in a text file, labels.txt, containing a list of labels provided by the User. Upon mmj2 unification, stepprover.c identified a label for each of the five theorems which 2-step proves it. For unisnALT 41552, the label list is a list of all 1-hypothesis propositional calculus deductions in set.mm. stepproverp.c is the same as stepprover.c except that it intermittently pauses during execution, allowing the User to observe the changes to a text file caused by the execution of particular statements of the program. (Contributed by Alan Sare, 19-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V        {𝐴} = 𝐴

20.36.11  Theorems with a proof in conventional notation derived from a VD proof

Theorems with a proof in conventional notation automatically derived by completeusersproof.c from a Virtual Deduction User's Proof.

TheoremnotnotrALT2 41553 Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)

TheoremsspwimpALT2 41554 If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Theoreme2ebindALT 41555 Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in e2ebindVD 41538. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))

Theoremax6e2ndALT 41556* If at least two sets exist (dtru 5258) , then the same is true expressed in an alternate form similar to the form of ax6e 2403. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in ax6e2ndVD 41534. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))

Theoremax6e2ndeqALT 41557* "At least two sets exist" expressed in the form of dtru 5258 is logically equivalent to the same expressed in a form similar to ax6e 2403 if dtru 5258 is false implies 𝑢 = 𝑣. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in ax6e2ndeqVD 41535. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))

Theorem2sb5ndALT 41558* Equivalence for double substitution 2sb5 2284 without distinct 𝑥, 𝑦 requirement. 2sb5nd 41186 is derived from 2sb5ndVD 41536. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in 2sb5ndVD 41536. (Contributed by Alan Sare, 19-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))

TheoremchordthmALT 41559* The intersecting chords theorem. If points A, B, C, and D lie on a circle (with center Q, say), and the point P is on the interior of the segments AB and CD, then the two products of lengths PA · PB and PC · PD are equal. The Euclidean plane is identified with the complex plane, and the fact that P is on AB and on CD is expressed by the hypothesis that the angles APB and CPD are equal to π. The result is proven by using chordthmlem5 25425 twice to show that PA · PB and PC · PD both equal BQ 2 PQ 2 . This is similar to the proof of the theorem given in Euclid's Elements, where it is Proposition III.35. Proven by David Moews on 28-Feb-2017 as chordthm 25426. https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 25426 is a Virtual Deduction User's Proof transcription of chordthm 25426. That VD User's Proof was input into completeusersproof, automatically generating this chordthmALT 41559 Metamath proof. (Contributed by Alan Sare, 19-Sep-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝑃 ∈ ℂ)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → ((𝐴𝑃)𝐹(𝐵𝑃)) = π)    &   (𝜑 → ((𝐶𝑃)𝐹(𝐷𝑃)) = π)    &   (𝜑𝑄 ∈ ℂ)    &   (𝜑 → (abs‘(𝐴𝑄)) = (abs‘(𝐵𝑄)))    &   (𝜑 → (abs‘(𝐴𝑄)) = (abs‘(𝐶𝑄)))    &   (𝜑 → (abs‘(𝐴𝑄)) = (abs‘(𝐷𝑄)))       (𝜑 → ((abs‘(𝑃𝐴)) · (abs‘(𝑃𝐵))) = ((abs‘(𝑃𝐶)) · (abs‘(𝑃𝐷))))

Theoremisosctrlem1ALT 41560 Lemma for isosctr 25410. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 25410. As it is verified by the Metamath program, isosctrlem1ALT 41560 verifies https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 41560. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)

Theoremiunconnlem2 41561* The indexed union of connected overlapping subspaces sharing a common point is connected. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/iunconlem2vd.html. As it is verified by the Metamath program, iunconnlem2 41561 verifies https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 41561. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 ↔ ((((((𝜑𝑢𝐽) ∧ 𝑣𝐽) ∧ (𝑢 𝑘𝐴 𝐵) ≠ ∅) ∧ (𝑣 𝑘𝐴 𝐵) ≠ ∅) ∧ (𝑢𝑣) ⊆ (𝑋 𝑘𝐴 𝐵)) ∧ 𝑘𝐴 𝐵 ⊆ (𝑢𝑣)))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝐵𝑋)    &   ((𝜑𝑘𝐴) → 𝑃𝐵)    &   ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Conn)       (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)

TheoremiunconnALT 41562* The indexed union of connected overlapping subspaces sharing a common point is connected. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/iunconaltvd.html. As it is verified by the Metamath program, iunconnALT 41562 verifies https://us.metamath.org/other/completeusersproof/iunconaltvd.html 41562. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝐵𝑋)    &   ((𝜑𝑘𝐴) → 𝑃𝐵)    &   ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Conn)       (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)

Theoremsineq0ALT 41563 A complex number whose sine is zero is an integer multiple of π. The Virtual Deduction form of the proof is https://us.metamath.org/other/completeusersproof/sineq0altvd.html. The Metamath form of the proof is sineq0ALT 41563. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 25119. The Virtual Deduction proof is verified by automatically transforming it into the Metamath form of the proof using completeusersproof, which is verified by the Metamath program. The proof of https://us.metamath.org/other/completeusersproof/sineq0altro.html 25119 is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ℂ → ((sin‘𝐴) = 0 ↔ (𝐴 / π) ∈ ℤ))

20.37  Mathbox for Glauco Siliprandi

20.37.1  Miscellanea

Theoremevth2f 41564* A version of evth2 23568 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑦𝐹    &   𝑥𝑋    &   𝑦𝑋    &   𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑥) ≤ (𝐹𝑦))

Theoremelunif 41565* A version of eluni 4827 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))

Theoremrzalf 41566 A version of rzal 4436 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥 𝐴 = ∅       (𝐴 = ∅ → ∀𝑥𝐴 𝜑)

Theoremfvelrnbf 41567 A version of fvelrnb 6717 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐹       (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))

Theoremrfcnpre1 41568 If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than a given extended real B is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐵    &   𝑥𝐹    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑋 = 𝐽    &   𝐴 = {𝑥𝑋𝐵 < (𝐹𝑥)}    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴𝐽)

Theoremubelsupr 41569* If U belongs to A and U is an upper bound, then U is the sup of A. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝐴 ⊆ ℝ ∧ 𝑈𝐴 ∧ ∀𝑥𝐴 𝑥𝑈) → 𝑈 = sup(𝐴, ℝ, < ))

Theoremfsumcnf 41570* A finite sum of functions to complex numbers from a common topological space is continuous, without disjoint var constraint x ph. The class expression for B normally contains free variables k and x to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))

Theoremmulltgt0 41571 The product of a negative and a positive number is negative. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 · 𝐵) < 0)

Theoremrspcegf 41572 A version of rspcev 3609 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝜓    &   𝑥𝐴    &   𝑥𝐵    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Theoremrabexgf 41573 A version of rabexg 5220 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremfcnre 41574 A function continuous with respect to the standard topology, is a real mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐹𝐶)       (𝜑𝐹:𝑇⟶ℝ)

Theoremsumsnd 41575* A sum of a singleton is the term. The deduction version of sumsn 15101. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑘𝐵)    &   𝑘𝜑    &   ((𝜑𝑘 = 𝑀) → 𝐴 = 𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremevthf 41576* A version of evth 23567 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑦𝐹    &   𝑥𝑋    &   𝑦𝑋    &   𝑥𝜑    &   𝑦𝜑    &   𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))

Theoremcnfex 41577 The class of continuous functions between two topologies is a set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

Theoremfnchoice 41578* For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝐴 ∈ Fin → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))

Theoremrefsumcn 41579* A finite sum of continuous real functions, from a common topological space, is continuous. The class expression for B normally contains free variables k and x to index it. See fsumcn 23478 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))

Theoremrfcnpre2 41580 If 𝐹 is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real 𝐵, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐵    &   𝑥𝐹    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑋 = 𝐽    &   𝐴 = {𝑥𝑋 ∣ (𝐹𝑥) < 𝐵}    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴𝐽)

Theoremcncmpmax 41581* When the hypothesis for the extreme value theorem hold, then the sup of the range of the function belongs to the range, it is real and it an upper bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑇 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑇 ≠ ∅)       (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡𝑇 (𝐹𝑡) ≤ sup(ran 𝐹, ℝ, < )))

Theoremrfcnpre3 41582* If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐴 = {𝑡𝑇𝐵 ≤ (𝐹𝑡)}    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴 ∈ (Clsd‘𝐽))

Theoremrfcnpre4 41583* If F is a continuous function with respect to the standard topology, then the preimage A of the values less than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐴 = {𝑡𝑇 ∣ (𝐹𝑡) ≤ 𝐵}    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴 ∈ (Clsd‘𝐽))

Theoremsumpair 41584* Sum of two distinct complex values. The class expression for 𝐴 and 𝐵 normally contain free variable 𝑘 to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑘𝐷)    &   (𝜑𝑘𝐸)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐸 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   ((𝜑𝑘 = 𝐴) → 𝐶 = 𝐷)    &   ((𝜑𝑘 = 𝐵) → 𝐶 = 𝐸)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))

Theoremrfcnnnub 41585* Given a real continuous function 𝐹 defined on a compact topological space, there is always a positive integer that is a strict upper bound of its range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   𝑇 = 𝐽    &   (𝜑𝑇 ≠ ∅)    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐹𝐶)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < 𝑛)

Theoremrefsum2cnlem1 41586* This is the core Lemma for refsum2cn 41587: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐺    &   𝑥𝜑    &   𝐴 = (𝑘 ∈ {1, 2} ↦ if(𝑘 = 1, 𝐹, 𝐺))    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ((𝐹𝑥) + (𝐺𝑥))) ∈ (𝐽 Cn 𝐾))

Theoremrefsum2cn 41587* The sum of two continuus real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑥𝐺    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ((𝐹𝑥) + (𝐺𝑥))) ∈ (𝐽 Cn 𝐾))

Theoremelunnel2 41588 A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐶) → 𝐴𝐵)

Theoremadantlllr 41589 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜑𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem3adantlr3 41590 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (((𝜑 ∧ (𝜓𝜒𝜂)) ∧ 𝜃) → 𝜏)

Theoremnnxrd 41591 A natural number is an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ*)

Theorem3adantll2 41592 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem3adantll3 41593 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremssnel 41594 If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)

Theoremelabrexg 41595* Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremsncldre 41596 A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ