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Mirrors > Home > ILE Home > Th. List > syl3anr2 | GIF version |
Description: A syllogism inference. (Contributed by NM, 1-Aug-2007.) |
Ref | Expression |
---|---|
syl3anr2.1 | ⊢ (𝜑 → 𝜃) |
syl3anr2.2 | ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂) |
Ref | Expression |
---|---|
syl3anr2 | ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑 ∧ 𝜏)) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anr2.1 | . . 3 ⊢ (𝜑 → 𝜃) | |
2 | syl3anr2.2 | . . . 4 ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂) | |
3 | 2 | ancoms 268 | . . 3 ⊢ (((𝜓 ∧ 𝜃 ∧ 𝜏) ∧ 𝜒) → 𝜂) |
4 | 1, 3 | syl3anl2 1297 | . 2 ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜏) ∧ 𝜒) → 𝜂) |
5 | 4 | ancoms 268 | 1 ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑 ∧ 𝜏)) → 𝜂) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 df-3an 981 |
This theorem is referenced by: mulgsubdir 13052 |
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