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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 216 | . 2 ⊢ (𝜑 → 𝜒) |
| 3 | syl3an2b.2 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | |
| 4 | 2, 3 | syl3an2 1165 | 1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 |
| 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 207 df-an 396 df-3an 1089 |
| This theorem is referenced by: omlimcl 8616 entrfil 9225 cflim2 10303 isdrngd 20765 isdrngdOLD 20767 rintopn 22915 cmpcld 23410 funvtxval0 29032 cusgr0v 29445 2clwwlk2clwwlklem 30365 cgrcomlr 35999 dissneqlem 37341 pmapglb 39772 |
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