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Theorem trsbcVD 44319
Description: 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 43982 is trsbcVD 44319 without virtual deductions and was automatically derived from trsbcVD 44319.
1:: (   đ´ ∈ 𝐵   â–ś   đ´ ∈ 𝐵   )
2:1: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)   )
3:1: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴)   )
4:1: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝐴)   )
5:1,2,3,4: (   đ´ ∈ 𝐵   â–ś   (([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))   )
6:1: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)))   )
7:5,6: (   đ´ ∈ 𝐵   â–ś   ([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: (   đ´ ∈ 𝐵   â–ś   ([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: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
13:9,12: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
14:13: (   đ´ ∈ 𝐵   â–ś   âˆ€đ‘Ś([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
15:14: (   đ´ ∈ 𝐵   â–ś   (∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
16:1: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
17:15,16: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
18:17: (   đ´ ∈ 𝐵   â–ś   âˆ€đ‘§([𝐴 / 𝑥]∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
19:18: (   đ´ ∈ 𝐵   â–ś   (∀𝑧[𝐴 / 𝑥]∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
20:1: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]∀𝑧∀𝑦(( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
21:19,20: (   đ´ ∈ 𝐵   â–ś   ([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: (   đ´ ∈ 𝐵   â–ś   ([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: (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]Tr 𝑥 ↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
27:23,26: (   đ´ ∈ 𝐵   â–ś   ([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.)
Assertion
Ref Expression
trsbcVD (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)
Proof of Theorem trsbcVD
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 44016 . . . . . . . . . . . . . 14 (   đ´ ∈ 𝐵   â–ś   đ´ ∈ 𝐵   )
2 sbcg 3855 . . . . . . . . . . . . . . 15 (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
31, 2e1a 44069 . . . . . . . . . . . . . 14 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)   )
4 sbcel2gv 3848 . . . . . . . . . . . . . . 15 (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴))
51, 4e1a 44069 . . . . . . . . . . . . . 14 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴)   )
6 sbcel2gv 3848 . . . . . . . . . . . . . . 15 (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝐴))
71, 6e1a 44069 . . . . . . . . . . . . . 14 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝐴)   )
8 imbi13 43962 . . . . . . . . . . . . . . 15 (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦) → (([𝐴 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴) → (([𝐴 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝐴) → (([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))))))
98a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ 𝐵 → (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦) → (([𝐴 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴) → (([𝐴 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝐴) → (([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))))))
101, 3, 5, 7, 9e1111 44117 . . . . . . . . . . . . 13 (   đ´ ∈ 𝐵   â–ś   (([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))   )
11 sbcim2g 43980 . . . . . . . . . . . . . 14 (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥))))
121, 11e1a 44069 . . . . . . . . . . . . 13 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)))   )
13 bibi1 350 . . . . . . . . . . . . . 14 (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥))) → (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))) ↔ (([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))))
1413biimprcd 249 . . . . . . . . . . . . 13 ((([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))) → (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 → ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥))) → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))))
1510, 12, 14e11 44130 . . . . . . . . . . . 12 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))   )
16 pm3.31 448 . . . . . . . . . . . . 13 ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)) → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
17 pm3.3 447 . . . . . . . . . . . . 13 (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
1816, 17impbii 208 . . . . . . . . . . . 12 ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
19 bibi1 350 . . . . . . . . . . . . 13 (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))) → (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) ↔ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))))
2019biimprd 247 . . . . . . . . . . . 12 (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))) → (((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))))
2115, 18, 20e10 44136 . . . . . . . . . . 11 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
22 pm3.31 448 . . . . . . . . . . . . . 14 ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
23 pm3.3 447 . . . . . . . . . . . . . 14 (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)))
2422, 23impbii 208 . . . . . . . . . . . . 13 ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
2524ax-gen 1789 . . . . . . . . . . . 12 ∀𝑥((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
26 sbcbi 43981 . . . . . . . . . . . 12 (𝐴 ∈ 𝐵 → (∀𝑥((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))))
271, 25, 26e10 44136 . . . . . . . . . . 11 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
28 bitr3 351 . . . . . . . . . . . 12 (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → ([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))))
2928com12 32 . . . . . . . . . . 11 (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → (([𝐴 / 𝑥](𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → ([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))))
3021, 27, 29e11 44130 . . . . . . . . . 10 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
3130gen11 44058 . . . . . . . . 9 (   đ´ ∈ 𝐵   â–ś   âˆ€đ‘Ś([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
32 albi 1812 . . . . . . . . 9 (∀𝑦([𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → (∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)))
3331, 32e1a 44069 . . . . . . . 8 (   đ´ ∈ 𝐵   â–ś   (∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
34 sbcal 3840 . . . . . . . . . 10 ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
3534a1i 11 . . . . . . . . 9 (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)))
361, 35e1a 44069 . . . . . . . 8 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
37 bibi1 350 . . . . . . . . 9 (([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → (([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) ↔ (∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))))
3837biimprcd 249 . . . . . . . 8 ((∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → (([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))))
3933, 36, 38e11 44130 . . . . . . 7 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
4039gen11 44058 . . . . . 6 (   đ´ ∈ 𝐵   â–ś   âˆ€đ‘§([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
41 albi 1812 . . . . . 6 (∀𝑧([𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → (∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)))
4240, 41e1a 44069 . . . . 5 (   đ´ ∈ 𝐵   â–ś   (∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
43 sbcal 3840 . . . . . . 7 ([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
4443a1i 11 . . . . . 6 (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)))
451, 44e1a 44069 . . . . 5 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
46 bibi1 350 . . . . . 6 (([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → (([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) ↔ (∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))))
4746biimprcd 249 . . . . 5 ((∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → (([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → ([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))))
4842, 45, 47e11 44130 . . . 4 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))   )
49 dftr2 5269 . . . 4 (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
50 biantr 804 . . . . 5 ((([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) ∧ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))) → ([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴))
5150ex 411 . . . 4 (([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → ((Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → ([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴)))
5248, 49, 51e10 44136 . . 3 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴)   )
53 dftr2 5269 . . . . 5 (Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
5453ax-gen 1789 . . . 4 ∀𝑥(Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
55 sbcbi 43981 . . . 4 (𝐴 ∈ 𝐵 → (∀𝑥(Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → ([𝐴 / 𝑥]Tr 𝑥 ↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))))
561, 54, 55e10 44136 . . 3 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]Tr 𝑥 ↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))   )
57 bibi1 350 . . . 4 (([𝐴 / 𝑥]Tr 𝑥 ↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → (([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴) ↔ ([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴)))
5857biimprcd 249 . . 3 (([𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴) → (([𝐴 / 𝑥]Tr 𝑥 ↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴)))
5952, 56, 58e11 44130 . 2 (   đ´ ∈ 𝐵   â–ś   ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴)   )
6059in1 44013 1 (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  âˆ€wal 1531   ∈ wcel 2098  [wsbc 3776  Tr wtr 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3473  df-sbc 3777  df-in 3954  df-ss 3964  df-uni 4911  df-tr 5268  df-vd1 44012
This theorem is referenced by: (None)
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