Theorem anbi12ci 457

Description: Variant of anbi12i 456 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 456	. 2 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))
4		ancom 264	. 2 ⊢ ((𝜓 ∧ 𝜃) ↔ (𝜃 ∧ 𝜓))
5	3, 4	bitri 183	1 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜃 ∧ 𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  opelopabsbALT  4237  cnvpom  5146  f1cnvcnv  5404