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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.) |
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 | syl3anl2.2 | . . . 4 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) | |
3 | 2 | ex 449 | . . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜏 → 𝜂)) |
4 | 1, 3 | syl3an2 1352 | . 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → (𝜏 → 𝜂)) |
5 | 4 | imp 444 | 1 ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 196 df-an 385 df-3an 1033 |
This theorem is referenced by: syl3anr2 1371 chfacfscmulcl 20481 chfacfscmulgsum 20484 chfacfpmmulcl 20485 chfacfpmmulgsum 20488 cpmadumatpolylem1 20505 cpmadumatpolylem2 20506 cpmadumatpoly 20507 chcoeffeqlem 20509 2atlt 33743 |
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