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Mirrors > Home > NFE Home > Th. List > an42 | 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 797 | . 2 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ((φ ∧ χ) ∧ (ψ ∧ θ))) | |
2 | ancom 437 | . . 3 ⊢ ((ψ ∧ θ) ↔ (θ ∧ ψ)) | |
3 | 2 | anbi2i 675 | . 2 ⊢ (((φ ∧ χ) ∧ (ψ ∧ θ)) ↔ ((φ ∧ χ) ∧ (θ ∧ ψ))) |
4 | 1, 3 | bitri 240 | 1 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ((φ ∧ χ) ∧ (θ ∧ ψ))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 |
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 177 df-an 360 |
This theorem is referenced by: rnlem 931 fnpprod 5843 |
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