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Mirrors > Home > MPE Home > Th. List > syl3anl2 | Structured version Visualization version GIF version |
Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2022.) |
Ref | Expression |
---|---|
syl3anl2.1 | ⊢ (𝜑 → 𝜒) |
syl3anl2.2 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
syl3anl2 | ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anl2.1 | . . 3 ⊢ (𝜑 → 𝜒) | |
2 | 1 | 3anim2i 1149 | . 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
3 | syl3anl2.2 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) | |
4 | 2, 3 | sylan 582 | 1 ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 |
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 209 df-an 399 df-3an 1085 |
This theorem is referenced by: chfacfscmulcl 21459 chfacfscmulgsum 21462 chfacfpmmulcl 21463 chfacfpmmulgsum 21466 cpmadumatpolylem1 21483 cpmadumatpolylem2 21484 cpmadumatpoly 21485 chcoeffeqlem 21487 2atlt 36569 |
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