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Mirrors > Home > MPE Home > Th. List > an42 | Structured version Visualization version GIF version |
Description: Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.) |
Ref | Expression |
---|---|
an42 | ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an4 654 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃))) | |
2 | ancom 463 | . . 3 ⊢ ((𝜓 ∧ 𝜃) ↔ (𝜃 ∧ 𝜓)) | |
3 | 2 | anbi2i 624 | . 2 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓))) |
4 | 1, 3 | bitri 277 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 |
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 209 df-an 399 |
This theorem is referenced by: an43 656 brecop2 8390 supmo 8915 infmo 8958 aceq1 9542 dfiso2 17041 eulerpartlemt0 31627 isbasisrelowllem1 34635 isbasisrelowllem2 34636 ifp1bi 39866 |
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