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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 608 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜃)) |
3 | sylanr1.2 | . 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏) | |
4 | 2, 3 | sylan2 586 | 1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 |
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 199 df-an 387 |
This theorem is referenced by: adantrll 713 adantrlr 714 sbthlem9 8353 pczpre 15930 cpmadugsumlemF 21058 blsscls2 22686 rpvmasumlem 25596 leopmuli 29543 chirredlem1 29800 chirredlem3 29802 dvconstbi 39372 bccbc 39383 reccot 43411 rectan 43412 aacllem 43457 |
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