Theorem bianim

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

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

Proof

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