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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ee112 37901 | e112 37900 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 37902 | 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 37903 | 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 37904 | e211 37903 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 37905 | 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 37906 | e210 37905 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 37907 | 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 37908 | e201 37907 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 37909 | 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 37910 | Virtual deduction rule e120 37909 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 37911 | 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 37912 | e021 37911 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 37913 | 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 37914 | e012 37913 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 37915 | 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 37916 | e102 37915 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 37917 | 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 37918 | Conjunction form of e22 37917. (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 37919 | e22an 37918 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 37920 | A virtual deduction elimination rule (see syl3c 64). (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 37921 | 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 37922 | 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 37923 | e110 37922 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 37924 | 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 37925 | e101 37924 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 37926 | 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 37927 | e011 37926 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 37928 | 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 37929 | e100 37928 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 37930 | 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 37931 | e010 37930 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 37932 | 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 37933 | e001 37932 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 37934 | 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 37935 | Conjunction form of e11 37934. (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 37936 | e11an 37935 without virtual deductions. syl22anc 1319 is also e11an 37935 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 37937 | 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 37938 | Conjunction form of e01 37937. (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 37939 | e01an 37938 without virtual deductions. sylancr 694 is also a form of e01an 37938 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 37940 | 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 37941 | Conjunction form of e10 37940. (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 37942 | e10an 37941 without virtual deductions. sylancl 693 is also e10an 37941 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 37943 | 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 37944 | Conjunction form of e02 37943. (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 37945 | e02an 37944 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 37946 | el021old 37947 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 37947 | 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 37948 | syl2an 493 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 37949 | 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 37950 | 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 37951 | 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 37952 | 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 37953 | 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 37954 | 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 37955 | 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 37956 | 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 37957 | 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 37958 | 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 37959 | 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 37960 | syl2an 493 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 37961 | syl2an 493 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 37962 | el0321old 37963 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 37963 | 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 37964 | el2122old 37965 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 37965 | 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 | eel0001 37966 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ (𝜃 → 𝜏) & ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜃 → 𝜂) | ||
Theorem | eel0000 37967 | Elimination rule similar to mp4an 705, 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 | eel1111 37968 | Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1322 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → 𝜏) & ⊢ ((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜑 → 𝜂) | ||
Theorem | eel00001 37969 | 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 37970 | Elimination rule similar eel0000 37967, 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 37971 | Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1330 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 37972 | A virtual deduction elimination rule (see sylsyld 59). (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 37973 | Conjunction form of e12 37972 (see syl6an 566). (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 37974 | Virtual deduction form of syl2an 493. (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 37975 | A virtual deduction elimination rule (see syl6mpi 65). (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 37976 | Conjunction form of e20 37975. (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 37977 | e20an 37976 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 37978 | A virtual deduction elimination rule (see syl6ci 69). (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 37979 | Conjunction form of e21 37978. (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 37980 | e21an 37979 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 37981 | 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 37982 | 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 37983 | Conjunction form of e33 37982. (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 37984 | e33an 37983 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 37985 | Meta-connective form of syl8 74. (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 37986 | Biconditional form of e3 37985. syl8ib 245 is e3bi 37986 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 37987 | Right biconditional form of e3 37985. (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 37988 | 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 37989 | e03 37988 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 37990 | Conjunction form of e03 37988. (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 37991 | Conjunction form of ee03 37989. (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 37992 | 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 37993 | e30 37992 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 37994 | 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 37995 | Conjunction form of ee30 37993. (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 37996 | 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 37997 | 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 37998 | e13an 37997 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 37999 | 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 38000 | e31 37999 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))) |
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