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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 4990 xmulpnf1 13313 odd2np1 16375 eltg3i 22984 restntr 23206 cmpcld 23426 rnelfm 23977 ovolctb2 25541 noextendseq 27727 iscgra 28832 isinag 28861 isleag 28870 iseqlg 28890 omlsilem 31431 mblfinlem3 37646 |
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