Theorem an42 798

Description: Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)

Assertion

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

Proof of Theorem an42

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)))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by:  rnlem  931  fnpprod  5844