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| Mirrors > Home > MPE Home > Th. List > sylanblc | Structured version Visualization version GIF version | ||
| Description: Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.) |
| Ref | Expression |
|---|---|
| sylanblc.1 | ⊢ (𝜑 → 𝜓) |
| sylanblc.2 | ⊢ 𝜒 |
| sylanblc.3 | ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜃) |
| Ref | Expression |
|---|---|
| sylanblc | ⊢ (𝜑 → 𝜃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sylanblc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
| 2 | sylanblc.2 | . 2 ⊢ 𝜒 | |
| 3 | sylanblc.3 | . . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜃) | |
| 4 | 3 | biimpi 216 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
| 5 | 1, 2, 4 | sylancl 586 | 1 ⊢ (𝜑 → 𝜃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 |
| 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 |
| This theorem is referenced by: uniintsn 4985 xmulpnf1 13316 odd2np1 16378 eltg3i 22968 restntr 23190 cmpcld 23410 rnelfm 23961 ovolctb2 25527 noextendseq 27712 iscgra 28817 isinag 28846 isleag 28855 iseqlg 28875 omlsilem 31421 mblfinlem3 37666 |
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