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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eel0TT 44701 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (⊤ → 𝜓) & ⊢ (⊤ → 𝜒) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ 𝜃 | ||
Theorem | eelT00 44702 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ 𝜃 | ||
Theorem | eelTTT 44703 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ (⊤ → 𝜓) & ⊢ (⊤ → 𝜒) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ 𝜃 | ||
Theorem | eelT11 44704 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ (𝜓 → 𝜒) & ⊢ (𝜓 → 𝜃) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜓 → 𝜏) | ||
Theorem | eelT1 44705 | Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Alan Sare, 23-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ (𝜓 → 𝜒) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜓 → 𝜃) | ||
Theorem | eelT12 44706 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ (𝜓 → 𝜒) & ⊢ (𝜃 → 𝜏) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜓 ∧ 𝜃) → 𝜂) | ||
Theorem | eelTT1 44707 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ (⊤ → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜒 → 𝜏) | ||
Theorem | eelT01 44708 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ 𝜓 & ⊢ (𝜒 → 𝜃) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜒 → 𝜏) | ||
Theorem | eel0T1 44709 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (⊤ → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜒 → 𝜏) | ||
Theorem | eel12131 44710 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ (𝜑 → 𝜓) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) & ⊢ ((𝜑 ∧ 𝜏) → 𝜂) & ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) | ||
Theorem | eel2131 44711 | syl2an 596 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜃) → 𝜏) & ⊢ ((𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜂) | ||
Theorem | eel3132 44712 | syl2an 596 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜃 ∧ 𝜓) → 𝜏) & ⊢ ((𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜂) | ||
Theorem | eel0321old 44713 | el0321old 44714 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) & ⊢ ((𝜑 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜂) | ||
Theorem | el0321old 44714 | A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( ( 𝜓 , 𝜒 , 𝜃 ) ▶ 𝜏 ) & ⊢ ((𝜑 ∧ 𝜏) → 𝜂) ⇒ ⊢ ( ( 𝜓 , 𝜒 , 𝜃 ) ▶ 𝜂 ) | ||
Theorem | eel2122old 44715 | el2122old 44716 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ (𝜓 → 𝜃) & ⊢ (𝜓 → 𝜏) & ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜂) | ||
Theorem | el2122old 44716 | A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝜒 ) & ⊢ ( 𝜓 ▶ 𝜃 ) & ⊢ ( 𝜓 ▶ 𝜏 ) & ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝜂 ) | ||
Theorem | eel0000 44717 | Elimination rule similar to mp4an 693, except with a left-nested conjunction unification theorem. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ 𝜃 & ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ 𝜏 | ||
Theorem | eel00001 44718 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ 𝜃 & ⊢ (𝜏 → 𝜂) & ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜁) ⇒ ⊢ (𝜏 → 𝜁) | ||
Theorem | eel00000 44719 | Elimination rule similar eel0000 44717, except with five hpothesis steps. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ 𝜃 & ⊢ 𝜏 & ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂) ⇒ ⊢ 𝜂 | ||
Theorem | eel11111 44720 | Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1381 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → 𝜏) & ⊢ (𝜑 → 𝜂) & ⊢ (((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁) ⇒ ⊢ (𝜑 → 𝜁) | ||
Theorem | e12 44721 | A virtual deduction elimination rule (see sylsyld 61). (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 , 𝜒 ▶ 𝜃 ) & ⊢ (𝜓 → (𝜃 → 𝜏)) ⇒ ⊢ ( 𝜑 , 𝜒 ▶ 𝜏 ) | ||
Theorem | e12an 44722 | Conjunction form of e12 44721 (see syl6an 684). (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 , 𝜒 ▶ 𝜃 ) & ⊢ ((𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ ( 𝜑 , 𝜒 ▶ 𝜏 ) | ||
Theorem | el12 44723 | Virtual deduction form of syl2an 596. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜏 ▶ 𝜒 ) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ( ( 𝜑 , 𝜏 ) ▶ 𝜃 ) | ||
Theorem | e20 44724 | A virtual deduction elimination rule (see syl6mpi 67). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ 𝜃 & ⊢ (𝜒 → (𝜃 → 𝜏)) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) | ||
Theorem | e20an 44725 | Conjunction form of e20 44724. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ 𝜃 & ⊢ ((𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) | ||
Theorem | ee20an 44726 | e20an 44725 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ 𝜃 & ⊢ ((𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜑 → (𝜓 → 𝜏)) | ||
Theorem | e21 44727 | A virtual deduction elimination rule (see syl6ci 71). (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 ▶ 𝜃 ) & ⊢ (𝜒 → (𝜃 → 𝜏)) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) | ||
Theorem | e21an 44728 | Conjunction form of e21 44727. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 ▶ 𝜃 ) & ⊢ ((𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) | ||
Theorem | ee21an 44729 | e21an 44728 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → 𝜃) & ⊢ ((𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜑 → (𝜓 → 𝜏)) | ||
Theorem | e333 44730 | A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜏 ) & ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) & ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁))) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜁 ) | ||
Theorem | e33 44731 | A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜏 ) & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | e33an 44732 | Conjunction form of e33 44731. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜏 ) & ⊢ ((𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee33an 44733 | e33an 44732 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) & ⊢ ((𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) | ||
Theorem | e3 44734 | Meta-connective form of syl8 76. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ (𝜃 → 𝜏) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜏 ) | ||
Theorem | e3bi 44735 | Biconditional form of e3 44734. syl8ib 256 is e3bi 44735 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ (𝜃 ↔ 𝜏) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜏 ) | ||
Theorem | e3bir 44736 | Right biconditional form of e3 44734. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ (𝜏 ↔ 𝜃) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜏 ) | ||
Theorem | e03 44737 | A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 , 𝜒 , 𝜃 ▶ 𝜏 ) & ⊢ (𝜑 → (𝜏 → 𝜂)) ⇒ ⊢ ( 𝜓 , 𝜒 , 𝜃 ▶ 𝜂 ) | ||
Theorem | ee03 44738 | e03 44737 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) & ⊢ (𝜑 → (𝜏 → 𝜂)) ⇒ ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜂))) | ||
Theorem | e03an 44739 | Conjunction form of e03 44737. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 , 𝜒 , 𝜃 ▶ 𝜏 ) & ⊢ ((𝜑 ∧ 𝜏) → 𝜂) ⇒ ⊢ ( 𝜓 , 𝜒 , 𝜃 ▶ 𝜂 ) | ||
Theorem | ee03an 44740 | Conjunction form of ee03 44738. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) & ⊢ ((𝜑 ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜂))) | ||
Theorem | e30 44741 | A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ 𝜏 & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee30 44742 | e30 44741 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ 𝜏 & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) | ||
Theorem | e30an 44743 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ 𝜏 & ⊢ ((𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee30an 44744 | Conjunction form of ee30 44742. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ 𝜏 & ⊢ ((𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) | ||
Theorem | e13 44745 | A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 , 𝜒 , 𝜃 ▶ 𝜏 ) & ⊢ (𝜓 → (𝜏 → 𝜂)) ⇒ ⊢ ( 𝜑 , 𝜒 , 𝜃 ▶ 𝜂 ) | ||
Theorem | e13an 44746 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 , 𝜒 , 𝜃 ▶ 𝜏 ) & ⊢ ((𝜓 ∧ 𝜏) → 𝜂) ⇒ ⊢ ( 𝜑 , 𝜒 , 𝜃 ▶ 𝜂 ) | ||
Theorem | ee13an 44747 | e13an 44746 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏))) & ⊢ ((𝜓 ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜂))) | ||
Theorem | e31 44748 | A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 ▶ 𝜏 ) & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee31 44749 | e31 44748 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → 𝜏) & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) | ||
Theorem | e31an 44750 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 ▶ 𝜏 ) & ⊢ ((𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee31an 44751 | e31an 44750 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → 𝜏) & ⊢ ((𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) | ||
Theorem | e23 44752 | A virtual deduction elimination rule (see syl10 79). (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜏 ) & ⊢ (𝜒 → (𝜏 → 𝜂)) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜂 ) | ||
Theorem | e23an 44753 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜏 ) & ⊢ ((𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜂 ) | ||
Theorem | ee23an 44754 | e23an 44753 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) & ⊢ ((𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂))) | ||
Theorem | e32 44755 | A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee32 44756 | e32 44755 without virtual deductions. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → 𝜏)) & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) | ||
Theorem | e32an 44757 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) & ⊢ ((𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee32an 44758 | e33an 44732 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → 𝜏)) & ⊢ ((𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) | ||
Theorem | e123 44759 | A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 , 𝜒 , 𝜏 ▶ 𝜂 ) & ⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁))) ⇒ ⊢ ( 𝜑 , 𝜒 , 𝜏 ▶ 𝜁 ) | ||
Theorem | ee123 44760 | e123 44759 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → (𝜒 → 𝜃)) & ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜂))) & ⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁))) ⇒ ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜁))) | ||
Theorem | el123 44761 | A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜏 ▶ 𝜂 ) & ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) ⇒ ⊢ ( ( 𝜑 , 𝜒 , 𝜏 ) ▶ 𝜁 ) | ||
Theorem | e233 44762 | A virtual deduction elimination rule. (Contributed by Alan Sare, 29-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜏 ) & ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜂 ) & ⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁))) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜁 ) | ||
Theorem | e323 44763 | A virtual deduction elimination rule. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) & ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜂 ) & ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁))) ⇒ ⊢ ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜁 ) | ||
Theorem | e000 44764 | A virtual deduction elimination rule. The non-virtual deduction form of e000 44764 is the virtual deduction form. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) ⇒ ⊢ 𝜃 | ||
Theorem | e00 44765 | Elimination rule identical to mp2 9. The non-virtual deduction form is the virtual deduction form, which is mp2 9. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ 𝜒 | ||
Theorem | e00an 44766 | Elimination rule identical to mp2an 692. The non-virtual deduction form is the virtual deduction form, which is mp2an 692. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ 𝜒 | ||
Theorem | eel00cT 44767 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (⊤ → 𝜒) | ||
Theorem | eelTT 44768 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ (⊤ → 𝜓) & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ 𝜒 | ||
Theorem | e0a 44769 | Elimination rule identical to ax-mp 5. The non-virtual deduction form is the virtual deduction form, which is ax-mp 5. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓 | ||
Theorem | eelT 44770 | An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓 | ||
Theorem | eel0cT 44771 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (⊤ → 𝜓) | ||
Theorem | eelT0 44772 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ 𝜒 | ||
Theorem | e0bi 44773 | Elimination rule identical to mpbi 230. The non-virtual deduction form is the virtual deduction form, which is mpbi 230. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ 𝜓 | ||
Theorem | e0bir 44774 | Elimination rule identical to mpbir 231. The non-virtual deduction form is the virtual deduction form, which is mpbir 231. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 ↔ 𝜑) ⇒ ⊢ 𝜓 | ||
Theorem | uun0.1 44775 | Convention notation form of un0.1 44776. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊤ → 𝜑) & ⊢ (𝜓 → 𝜒) & ⊢ ((⊤ ∧ 𝜓) → 𝜃) ⇒ ⊢ (𝜓 → 𝜃) | ||
Theorem | un0.1 44776 | ⊤ is the constant true, a tautology (see df-tru 1539). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( ⊤ ▶ 𝜑 ) & ⊢ ( 𝜓 ▶ 𝜒 ) & ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) ⇒ ⊢ ( 𝜓 ▶ 𝜃 ) | ||
Theorem | uunT1 44777 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to accommodate a possible future version of df-tru 1539. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((⊤ ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | uunT1p1 44778 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ ⊤) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | uunT21 44779 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((⊤ ∧ (𝜑 ∧ 𝜓)) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | uun121 44780 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | uun121p1 44781 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | uun132 44782 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | uun132p1 44783 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | anabss7p1 44784 | A deduction unionizing a non-unionized collection of virtual hypotheses. This would have been named uun221 if the zeroth permutation did not exist in set.mm as anabss7 673. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜒) | ||
Theorem | un10 44785 | A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( ( 𝜑 , ⊤ ) ▶ 𝜓 ) ⇒ ⊢ ( 𝜑 ▶ 𝜓 ) | ||
Theorem | un01 44786 | A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( ( ⊤ , 𝜑 ) ▶ 𝜓 ) ⇒ ⊢ ( 𝜑 ▶ 𝜓 ) | ||
Theorem | un2122 44787 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | uun2131 44788 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | uun2131p1 44789 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜓)) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | uunTT1 44790 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((⊤ ∧ ⊤ ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | uunTT1p1 44791 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((⊤ ∧ 𝜑 ∧ ⊤) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | uunTT1p2 44792 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ ⊤ ∧ ⊤) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | uunT11 44793 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((⊤ ∧ 𝜑 ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | uunT11p1 44794 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ ⊤ ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | uunT11p2 44795 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜑 ∧ ⊤) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | uunT12 44796 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((⊤ ∧ 𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | uunT12p1 44797 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((⊤ ∧ 𝜓 ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | uunT12p2 44798 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | uunT12p3 44799 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | uunT12p4 44800 | A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ ⊤) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
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