Theorem anbi12ci 679

Description: Variant of anbi12i 678 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
anbi12.1	⊢ (φ ↔ ψ)
anbi12.2	⊢ (χ ↔ θ)

Assertion

Ref	Expression
anbi12ci	⊢ ((φ ∧ χ) ↔ (θ ∧ ψ))

Proof of Theorem anbi12ci

Step	Hyp	Ref	Expression
1		anbi12.1	. . 3 ⊢ (φ ↔ ψ)
2		anbi12.2	. . 3 ⊢ (χ ↔ θ)
3	1, 2	anbi12i 678	. 2 ⊢ ((φ ∧ χ) ↔ (ψ ∧ θ))
4		ancom 437	. 2 ⊢ ((ψ ∧ θ) ↔ (θ ∧ ψ))
5	3, 4	bitri 240	1 ⊢ ((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:  funsex  5829  refex  5912  foundex  5915  csucex  6260