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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 206 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
5 | 1, 2, 4 | sylancl 694 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 |
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 197 df-an 386 |
This theorem is referenced by: odd2np1 15065 restntr 20986 cmpcld 21205 rnelfm 21757 ovolctb2 23260 omlsilem 28261 noextendseq 31820 mblfinlem3 33448 |
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