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Mirrors > Home > ILE Home > Th. List > syl113anc | GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
Ref | Expression |
---|---|
sylXanc.1 | ⊢ (𝜑 → 𝜓) |
sylXanc.2 | ⊢ (𝜑 → 𝜒) |
sylXanc.3 | ⊢ (𝜑 → 𝜃) |
sylXanc.4 | ⊢ (𝜑 → 𝜏) |
sylXanc.5 | ⊢ (𝜑 → 𝜂) |
syl113anc.6 | ⊢ ((𝜓 ∧ 𝜒 ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) → 𝜁) |
Ref | Expression |
---|---|
syl113anc | ⊢ (𝜑 → 𝜁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylXanc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | sylXanc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | sylXanc.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
4 | sylXanc.4 | . . 3 ⊢ (𝜑 → 𝜏) | |
5 | sylXanc.5 | . . 3 ⊢ (𝜑 → 𝜂) | |
6 | 3, 4, 5 | 3jca 1172 | . 2 ⊢ (𝜑 → (𝜃 ∧ 𝜏 ∧ 𝜂)) |
7 | syl113anc.6 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) → 𝜁) | |
8 | 1, 2, 6, 7 | syl3anc 1233 | 1 ⊢ (𝜑 → 𝜁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 975 |
This theorem is referenced by: syl123anc 1250 syl213anc 1252 divalglemnn 11870 pythagtriplem18 12228 |
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