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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ee011 44701 | e011 44700 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 44702 | 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 44703 | e100 44702 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 44704 | 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 44705 | e010 44704 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 44706 | 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 44707 | e001 44706 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 44708 | 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 44709 | Conjunction form of e11 44708. (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 44710 | e11an 44709 without virtual deductions. syl22anc 839 is also e11an 44709 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 44711 | 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 44712 | Conjunction form of e01 44711. (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 44713 | e01an 44712 without virtual deductions. sylancr 587 is also a form of e01an 44712 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 44714 | 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 44715 | Conjunction form of e10 44714. (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 44716 | e10an 44715 without virtual deductions. sylancl 586 is also e10an 44715 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 44717 | 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 44718 | Conjunction form of e02 44717. (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 44719 | e02an 44718 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 44720 | el021old 44721 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 44721 | 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 44722 | syl2an 596 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 44723 | 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 44724 | 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 44725 | 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 44726 | 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 44727 | 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 44728 | 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 44729 | 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 44730 | 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 44731 | 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 44732 | 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 44733 | 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 44734 | 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 44735 | 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 44736 | el0321old 44737 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 44737 | 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 44738 | el2122old 44739 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 44739 | 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 44740 | 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 44741 | 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 44742 | Elimination rule similar eel0000 44740, 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 44743 | Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1384 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 44744 | 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 44745 | Conjunction form of e12 44744 (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 44746 | 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 44747 | 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 44748 | Conjunction form of e20 44747. (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 44749 | e20an 44748 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 44750 | 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 44751 | Conjunction form of e21 44750. (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 44752 | e21an 44751 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 44753 | 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 44754 | 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 44755 | Conjunction form of e33 44754. (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 44756 | e33an 44755 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 44757 | 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 44758 | Biconditional form of e3 44757. syl8ib 256 is e3bi 44758 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 44759 | Right biconditional form of e3 44757. (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 44760 | 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 44761 | e03 44760 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 44762 | Conjunction form of e03 44760. (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 44763 | Conjunction form of ee03 44761. (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 44764 | 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 44765 | e30 44764 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 44766 | 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 44767 | Conjunction form of ee30 44765. (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 44768 | 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 44769 | 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 44770 | e13an 44769 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 44771 | 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 44772 | e31 44771 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 44773 | 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 44774 | e31an 44773 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 44775 | 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 44776 | 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 44777 | e23an 44776 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 44778 | 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 44779 | e32 44778 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 44780 | 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 44781 | e33an 44755 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 44782 | 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 44783 | e123 44782 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 44784 | 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 44785 | 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 44786 | 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 44787 | A virtual deduction elimination rule. The non-virtual deduction form of e000 44787 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 44788 | 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 44789 | 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 44790 | 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 44791 | 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 44792 | 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 44793 | An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (⊤ → 𝜑) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓 | ||
| Theorem | eel0cT 44794 | An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (⊤ → 𝜓) | ||
| Theorem | eelT0 44795 | 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 44796 | 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 44797 | 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 44798 | Convention notation form of un0.1 44799. (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 44799 | ⊤ is the constant true, a tautology (see df-tru 1543). 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 44800 | 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 1543. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ((⊤ ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
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