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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 617 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜃)) |
3 | sylanr1.2 | . 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏) | |
4 | 2, 3 | sylan2 595 | 1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 |
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 210 df-an 400 |
This theorem is referenced by: adantrll 721 adantrlr 722 sbthlem9 8619 pczpre 16174 cpmadugsumlemF 21481 blsscls2 23111 rpvmasumlem 26071 leopmuli 29916 chirredlem1 30173 chirredlem3 30175 pibt2 34834 dvconstbi 41038 bccbc 41049 reccot 45284 rectan 45285 aacllem 45329 |
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