![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 215 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
5 | 1, 2, 4 | sylancl 587 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 |
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 206 df-an 398 |
This theorem is referenced by: uniintsn 4992 xmulpnf1 13253 odd2np1 16284 eltg3i 22464 restntr 22686 cmpcld 22906 rnelfm 23457 ovolctb2 25009 noextendseq 27170 iscgra 28060 isinag 28089 isleag 28098 iseqlg 28118 omlsilem 30655 mblfinlem3 36527 |
Copyright terms: Public domain | W3C validator |