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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | e220 43701 | 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 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) | ||
Theorem | ee220 43702 | e220 43701 without virtual deductions. (Contributed by Alan Sare, 12-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ 𝜏 & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜂)) | ||
Theorem | e202 43703 | 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 ) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) | ||
Theorem | ee202 43704 | e202 43703 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ 𝜃 & ⊢ (𝜑 → (𝜓 → 𝜏)) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜂)) | ||
Theorem | e022 43705 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜓 , 𝜒 ▶ 𝜏 ) & ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee022 43706 | e022 43705 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → (𝜒 → 𝜃)) & ⊢ (𝜓 → (𝜒 → 𝜏)) & ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜓 → (𝜒 → 𝜂)) | ||
Theorem | e002 43707 | 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 ) & ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜒 , 𝜃 ▶ 𝜂 ) | ||
Theorem | ee002 43708 | e002 43707 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ (𝜒 → (𝜃 → 𝜏)) & ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜒 → (𝜃 → 𝜂)) | ||
Theorem | e020 43709 | 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 & ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee020 43710 | e020 43709 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → (𝜒 → 𝜃)) & ⊢ 𝜏 & ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜓 → (𝜒 → 𝜂)) | ||
Theorem | e200 43711 | 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 & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) | ||
Theorem | ee200 43712 | e200 43711 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ 𝜃 & ⊢ 𝜏 & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜂)) | ||
Theorem | e221 43713 | 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 ) & ⊢ ( 𝜑 ▶ 𝜏 ) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) | ||
Theorem | ee221 43714 | e221 43713 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → 𝜏) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜂)) | ||
Theorem | e212 43715 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 ▶ 𝜃 ) & ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) | ||
Theorem | ee212 43716 | e212 43715 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → (𝜓 → 𝜏)) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜂)) | ||
Theorem | e122 43717 | 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 ) & ⊢ ( 𝜑 , 𝜒 ▶ 𝜏 ) & ⊢ (𝜓 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜒 ▶ 𝜂 ) | ||
Theorem | e112 43718 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 ▶ 𝜒 ) & ⊢ ( 𝜑 , 𝜃 ▶ 𝜏 ) & ⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜃 ▶ 𝜂 ) | ||
Theorem | ee112 43719 | e112 43718 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → (𝜃 → 𝜏)) & ⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜃 → 𝜂)) | ||
Theorem | e121 43720 | 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 ) & ⊢ ( 𝜑 ▶ 𝜏 ) & ⊢ (𝜓 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜒 ▶ 𝜂 ) | ||
Theorem | e211 43721 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 ▶ 𝜃 ) & ⊢ ( 𝜑 ▶ 𝜏 ) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) | ||
Theorem | ee211 43722 | e211 43721 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → 𝜏) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜂)) | ||
Theorem | e210 43723 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 ▶ 𝜃 ) & ⊢ 𝜏 & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) | ||
Theorem | ee210 43724 | e210 43723 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → 𝜃) & ⊢ 𝜏 & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜂)) | ||
Theorem | e201 43725 | 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 ) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) | ||
Theorem | ee201 43726 | e201 43725 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ 𝜃 & ⊢ (𝜑 → 𝜏) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜂)) | ||
Theorem | e120 43727 | A virtual deduction elimination rule. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 , 𝜒 ▶ 𝜃 ) & ⊢ 𝜏 & ⊢ (𝜓 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee120 43728 | Virtual deduction rule e120 43727 without virtual deduction symbols. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → (𝜒 → 𝜃)) & ⊢ 𝜏 & ⊢ (𝜓 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜒 → 𝜂)) | ||
Theorem | e021 43729 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ( 𝜓 ▶ 𝜏 ) & ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜓 , 𝜒 ▶ 𝜂 ) | ||
Theorem | ee021 43730 | e021 43729 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → (𝜒 → 𝜃)) & ⊢ (𝜓 → 𝜏) & ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜓 → (𝜒 → 𝜂)) | ||
Theorem | e012 43731 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜓 , 𝜃 ▶ 𝜏 ) & ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜓 , 𝜃 ▶ 𝜂 ) | ||
Theorem | ee012 43732 | e012 43731 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ (𝜓 → (𝜃 → 𝜏)) & ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜓 → (𝜃 → 𝜂)) | ||
Theorem | e102 43733 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ 𝜒 & ⊢ ( 𝜑 , 𝜃 ▶ 𝜏 ) & ⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂))) ⇒ ⊢ ( 𝜑 , 𝜃 ▶ 𝜂 ) | ||
Theorem | ee102 43734 | e102 43733 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ 𝜒 & ⊢ (𝜑 → (𝜃 → 𝜏)) & ⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜃 → 𝜂)) | ||
Theorem | e22 43735 | A virtual deduction elimination rule. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 , 𝜓 ▶ 𝜃 ) & ⊢ (𝜒 → (𝜃 → 𝜏)) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) | ||
Theorem | e22an 43736 | Conjunction form of e22 43735. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜑 , 𝜓 ▶ 𝜃 ) & ⊢ ((𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) | ||
Theorem | ee22an 43737 | e22an 43736 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ ((𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜑 → (𝜓 → 𝜏)) | ||
Theorem | e111 43738 | A virtual deduction elimination rule (see syl3c 66). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 ▶ 𝜒 ) & ⊢ ( 𝜑 ▶ 𝜃 ) & ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ ( 𝜑 ▶ 𝜏 ) | ||
Theorem | e1111 43739 | A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 ▶ 𝜒 ) & ⊢ ( 𝜑 ▶ 𝜃 ) & ⊢ ( 𝜑 ▶ 𝜏 ) & ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))) ⇒ ⊢ ( 𝜑 ▶ 𝜂 ) | ||
Theorem | e110 43740 | 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 & ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ ( 𝜑 ▶ 𝜏 ) | ||
Theorem | ee110 43741 | e110 43740 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ 𝜃 & ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | e101 43742 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ 𝜒 & ⊢ ( 𝜑 ▶ 𝜃 ) & ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ ( 𝜑 ▶ 𝜏 ) | ||
Theorem | ee101 43743 | e101 43742 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ 𝜒 & ⊢ (𝜑 → 𝜃) & ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | e011 43744 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 ▶ 𝜒 ) & ⊢ ( 𝜓 ▶ 𝜃 ) & ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ ( 𝜓 ▶ 𝜏 ) | ||
Theorem | ee011 43745 | e011 43744 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ (𝜓 → 𝜃) & ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ (𝜓 → 𝜏) | ||
Theorem | e100 43746 | A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ 𝜒 & ⊢ 𝜃 & ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ ( 𝜑 ▶ 𝜏 ) | ||
Theorem | ee100 43747 | e100 43746 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ 𝜒 & ⊢ 𝜃 & ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | e010 43748 | 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 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ ( 𝜓 ▶ 𝜏 ) | ||
Theorem | ee010 43749 | e010 43748 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ 𝜃 & ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏))) ⇒ ⊢ (𝜓 → 𝜏) | ||
Theorem | e001 43750 | 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 → (𝜃 → 𝜏))) ⇒ ⊢ ( 𝜒 ▶ 𝜏 ) | ||
Theorem | ee001 43751 | e001 43750 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ (𝜒 → 𝜃) & ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) ⇒ ⊢ (𝜒 → 𝜏) | ||
Theorem | e11 43752 | A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 ▶ 𝜒 ) & ⊢ (𝜓 → (𝜒 → 𝜃)) ⇒ ⊢ ( 𝜑 ▶ 𝜃 ) | ||
Theorem | e11an 43753 | Conjunction form of e11 43752. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ ( 𝜑 ▶ 𝜒 ) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ( 𝜑 ▶ 𝜃 ) | ||
Theorem | ee11an 43754 | e11an 43753 without virtual deductions. syl22anc 836 is also e11an 43753 without virtual deductions, exept with a different order of hypotheses. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | e01 43755 | A virtual deduction elimination rule. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 ▶ 𝜒 ) & ⊢ (𝜑 → (𝜒 → 𝜃)) ⇒ ⊢ ( 𝜓 ▶ 𝜃 ) | ||
Theorem | e01an 43756 | Conjunction form of e01 43755. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 ▶ 𝜒 ) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ( 𝜓 ▶ 𝜃 ) | ||
Theorem | ee01an 43757 | e01an 43756 without virtual deductions. sylancr 586 is also a form of e01an 43756 without virtual deduction, except the order of the hypotheses is different. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜓 → 𝜃) | ||
Theorem | e10 43758 | A virtual deduction elimination rule (see mpisyl 21). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ 𝜒 & ⊢ (𝜓 → (𝜒 → 𝜃)) ⇒ ⊢ ( 𝜑 ▶ 𝜃 ) | ||
Theorem | e10an 43759 | Conjunction form of e10 43758. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( 𝜑 ▶ 𝜓 ) & ⊢ 𝜒 & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ( 𝜑 ▶ 𝜃 ) | ||
Theorem | ee10an 43760 | e10an 43759 without virtual deductions. sylancl 585 is also e10an 43759 without virtual deductions, except the order of the hypotheses is different. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ 𝜒 & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | e02 43761 | A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ (𝜑 → (𝜃 → 𝜏)) ⇒ ⊢ ( 𝜓 , 𝜒 ▶ 𝜏 ) | ||
Theorem | e02an 43762 | Conjunction form of e02 43761. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( 𝜓 , 𝜒 ▶ 𝜃 ) & ⊢ ((𝜑 ∧ 𝜃) → 𝜏) ⇒ ⊢ ( 𝜓 , 𝜒 ▶ 𝜏 ) | ||
Theorem | ee02an 43763 | e02an 43762 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜓 → (𝜒 → 𝜃)) & ⊢ ((𝜑 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜓 → (𝜒 → 𝜏)) | ||
Theorem | eel021old 43764 | el021old 43765 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) & ⊢ ((𝜑 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜓 ∧ 𝜒) → 𝜏) | ||
Theorem | el021old 43765 | A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ ( ( 𝜓 , 𝜒 ) ▶ 𝜃 ) & ⊢ ((𝜑 ∧ 𝜃) → 𝜏) ⇒ ⊢ ( ( 𝜓 , 𝜒 ) ▶ 𝜏 ) | ||
Theorem | eel132 43766 | syl2an 595 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.) |
⊢ (𝜑 → 𝜓) & ⊢ ((𝜒 ∧ 𝜃) → 𝜏) & ⊢ ((𝜓 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂) | ||
Theorem | eel000cT 43767 | 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 | eel0TT 43768 | 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 43769 | 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 43770 | 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 43771 | 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 43772 | 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 43773 | 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 43774 | 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 43775 | 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 43776 | 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 43777 | 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 43778 | syl2an 595 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 43779 | syl2an 595 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 43780 | el0321old 43781 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 43781 | 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 43782 | el2122old 43783 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 43783 | 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 43784 | Elimination rule similar to mp4an 690, 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 43785 | 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 43786 | Elimination rule similar eel0000 43784, 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 43787 | 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 43788 | 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 43789 | Conjunction form of e12 43788 (see syl6an 681). (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 43790 | Virtual deduction form of syl2an 595. (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 43791 | 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 43792 | Conjunction form of e20 43791. (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 43793 | e20an 43792 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 43794 | 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 43795 | Conjunction form of e21 43794. (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 43796 | e21an 43795 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 43797 | 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 43798 | 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 43799 | Conjunction form of e33 43798. (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 43800 | e33an 43799 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))) |
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