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Mirrors > Home > MPE Home > Th. List > sylanr1 | Structured version Visualization version GIF version |
Description: A syllogism inference. (Contributed by NM, 9-Apr-2005.) |
Ref | Expression |
---|---|
sylanr1.1 | ⊢ (𝜑 → 𝜒) |
sylanr1.2 | ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏) |
Ref | Expression |
---|---|
sylanr1 | ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylanr1.1 | . . 3 ⊢ (𝜑 → 𝜒) | |
2 | 1 | anim1i 613 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜃)) |
3 | sylanr1.2 | . 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏) | |
4 | 2, 3 | sylan2 591 | 1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 395 |
This theorem is referenced by: adantrll 720 adantrlr 721 sbthlem9 9118 unfi 9199 pczpre 16841 cpmadugsumlemF 22863 blsscls2 24498 rpvmasumlem 27510 leopmuli 32060 chirredlem1 32317 chirredlem3 32319 pibt2 37134 mhpind 42281 dvconstbi 44042 bccbc 44053 reccot 48537 rectan 48538 aacllem 48582 |
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