Theorem an42 389

Description: Rearrangement of 4 conjuncts.

Assertion

Ref	Expression
an42	⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ((φ ∧ χ) ∧ (θ ∧ ψ)))

Proof of Theorem an42

Step	Hyp	Ref	Expression
1		an4 388	. 2 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ((φ ∧ χ) ∧ (ψ ∧ θ)))
2		ancom 333	. . 3 ⊢ ((ψ ∧ θ) ↔ (θ ∧ ψ))
3	2	anbi2i 367	. 2 ⊢ (((φ ∧ χ) ∧ (ψ ∧ θ)) ↔ ((φ ∧ χ) ∧ (θ ∧ ψ)))
4	1, 3	bitr 151	1 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ((φ ∧ χ) ∧ (θ ∧ ψ)))

Syntax hints:   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  an42s 391  pssn2lp 1571  brecop2 3243  aceq1 3552  prlem934b 3932  prlem934 3933

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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