Theorem pm5.32dav

Description: Distribution of implication over biconditional (deduction form). Variant of pm5.32da . (Contributed by Zhi Wang, 30-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		pm5.32dav.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) )
2	1	pm5.32da	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) ) )
3		ancom	⊢ ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜒 ∧ 𝜓 ) )
4		ancom	⊢ ( ( 𝜓 ∧ 𝜃 ) ↔ ( 𝜃 ∧ 𝜓 ) )
5	2 3 4	3bitr3g	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜃 ∧ 𝜓 ) ) )