Theorem bianim 586

Description: Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)

Hypotheses

Ref	Expression
bianim.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
bianim.2	⊢ (𝜒 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
bianim	⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))

Proof of Theorem bianim

Step	Hyp	Ref	Expression
1		bianim.1	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2		bianim.2	. . 3 ⊢ (𝜒 → (𝜓 ↔ 𝜃))
3	2	pm5.32ri 585	. 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜃 ∧ 𝜒))
4	1, 3	bitri 278	1 ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400

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

This proof depends on definitions:  df-bi 210  df-an 401

This theorem is used by:  dfcnvrefrel4  39289  dfcnvrefrel5  39290  stgredgel  48750  dfidom2  49136