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Mirrors > Home > MPE Home > Th. List > an12s | Structured version Visualization version GIF version |
Description: Swap two conjuncts in antecedent. The label suffix "s" means that an12 644 is combined with syl 17 (or a variant). (Contributed by NM, 13-Mar-1996.) |
Ref | Expression |
---|---|
an12s.1 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
Ref | Expression |
---|---|
an12s | ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜒)) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an12 644 | . 2 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜒)) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒))) | |
2 | an12s.1 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜒)) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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: anabsan2 673 oecl 8537 oaass 8561 odi 8579 oen0 8586 oeworde 8593 ltexprlem4 11034 iccshftr 13463 iccshftl 13465 iccdil 13467 icccntr 13469 ndvdsadd 16353 eulerthlem2 16715 neips 22617 tx1stc 23154 filuni 23389 ufldom 23466 isch3 30494 unoplin 31173 hmoplin 31195 adjlnop 31339 chirredlem2 31644 btwnconn1lem12 35070 btwnconn1 35073 finxpreclem2 36271 poimirlem25 36513 mblfinlem4 36528 iscringd 36866 unichnidl 36899 |
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