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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ee210 41001 | e210 41000 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 41002 | 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 41003 | e201 41002 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 41004 | 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 41005 | Virtual deduction rule e120 41004 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 41006 | 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 41007 | e021 41006 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 41008 | 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 41009 | e012 41008 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 41010 | 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 41011 | e102 41010 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 41012 | 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 41013 | Conjunction form of e22 41012. (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 41014 | e22an 41013 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 41015 | 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 41016 | 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 41017 | 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 41018 | e110 41017 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 41019 | 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 41020 | e101 41019 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 41021 | 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 41022 | e011 41021 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 41023 | 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 41024 | e100 41023 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 41025 | 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 41026 | e010 41025 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 41027 | 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 41028 | e001 41027 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 41029 | 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 41030 | Conjunction form of e11 41029. (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 41031 | e11an 41030 without virtual deductions. syl22anc 836 is also e11an 41030 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 41032 | 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 41033 | Conjunction form of e01 41032. (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 41034 | e01an 41033 without virtual deductions. sylancr 589 is also a form of e01an 41033 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 41035 | 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 41036 | Conjunction form of e10 41035. (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 41037 | e10an 41036 without virtual deductions. sylancl 588 is also e10an 41036 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 41038 | 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 41039 | Conjunction form of e02 41038. (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 41040 | e02an 41039 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 41041 | el021old 41042 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 41042 | 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 41043 | syl2an 597 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 41044 | 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 41045 | 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 41046 | 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 41047 | 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 41048 | 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 41049 | 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 41050 | 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 41051 | 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 41052 | 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 41053 | 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 41054 | 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 41055 | syl2an 597 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 41056 | syl2an 597 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 41057 | el0321old 41058 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 41058 | 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 41059 | el2122old 41060 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 41060 | 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 41061 | Elimination rule similar to mp4an 691, 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 41062 | 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 41063 | Elimination rule similar eel0000 41061, 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 41064 | Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1378 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 41065 | 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 41066 | Conjunction form of e12 41065 (see syl6an 682). (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 41067 | Virtual deduction form of syl2an 597. (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 41068 | 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 41069 | Conjunction form of e20 41068. (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 41070 | e20an 41069 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 41071 | 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 41072 | Conjunction form of e21 41071. (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 41073 | e21an 41072 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 41074 | 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 41075 | 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 41076 | Conjunction form of e33 41075. (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 41077 | e33an 41076 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 41078 | 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 41079 | Biconditional form of e3 41078. syl8ib 258 is e3bi 41079 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 41080 | Right biconditional form of e3 41078. (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 41081 | 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 41082 | e03 41081 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 41083 | Conjunction form of e03 41081. (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 41084 | Conjunction form of ee03 41082. (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 41085 | 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 41086 | e30 41085 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 41087 | 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 41088 | Conjunction form of ee30 41086. (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 41089 | 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 41090 | 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 41091 | e13an 41090 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 41092 | 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 41093 | e31 41092 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 41094 | 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 41095 | e31an 41094 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 41096 | 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 41097 | 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 41098 | e23an 41097 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 41099 | 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 41100 | e32 41099 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))) |
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