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Mirrors > Home > MPE Home > Th. List > syl3an2b | Structured version Visualization version GIF version |
Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
Ref | Expression |
---|---|
syl3an2b.1 | ⊢ (𝜑 ↔ 𝜒) |
syl3an2b.2 | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syl3an2b | ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3an2b.1 | . . 3 ⊢ (𝜑 ↔ 𝜒) | |
2 | 1 | biimpi 219 | . 2 ⊢ (𝜑 → 𝜒) |
3 | syl3an2b.2 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | |
4 | 2, 3 | syl3an2 1166 | 1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1089 |
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 df-3an 1091 |
This theorem is referenced by: omlimcl 8306 entrfil 8863 cflim2 9877 isdrngd 19792 rintopn 21806 cmpcld 22299 funvtxval0 27106 cusgr0v 27516 2clwwlk2clwwlklem 28429 cgrcomlr 34037 dissneqlem 35248 pmapglb 37521 |
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