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Mirrors > Home > ILE Home > Th. List > syl222anc | 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) |
sylXanc.6 | ⊢ (𝜑 → 𝜁) |
syl222anc.7 | ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → 𝜎) |
Ref | Expression |
---|---|
syl222anc | ⊢ (𝜑 → 𝜎) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylXanc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | sylXanc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | sylXanc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
4 | sylXanc.4 | . 2 ⊢ (𝜑 → 𝜏) | |
5 | sylXanc.5 | . . 3 ⊢ (𝜑 → 𝜂) | |
6 | sylXanc.6 | . . 3 ⊢ (𝜑 → 𝜁) | |
7 | 5, 6 | jca 304 | . 2 ⊢ (𝜑 → (𝜂 ∧ 𝜁)) |
8 | syl222anc.7 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → 𝜎) | |
9 | 1, 2, 3, 4, 7, 8 | syl221anc 1244 | 1 ⊢ (𝜑 → 𝜎) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ 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: 3anandis 1342 3anandirs 1343 xaddass2 9814 xpncan 9815 divdenle 12138 pockthlem 12295 bdxmet 13216 |
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