Theorem pm5.32ri

Description: Distribution of implication over biconditional (inference form). (Contributed by NM, 12-Mar-1995)

	Ref	Expression
Hypothesis	pm5.32i.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	pm5.32ri	⊢ ( ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜒 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.32i.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	pm5.32i	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) )
3		ancom	⊢ ( ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) )
4		ancom	⊢ ( ( 𝜒 ∧ 𝜑 ) ↔ ( 𝜑 ∧ 𝜒 ) )
5	2 3 4	3bitr4i	⊢ ( ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜒 ∧ 𝜑 ) )