Theorem bianim 576

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 575	. 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜃 ∧ 𝜒))
4	1, 3	bitri 275	1 ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  dfcnvrefrel4  38492  dfcnvrefrel5  38493  stgredgel  47882