Theorem pm5.32da

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006)

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

Proof

Step	Hyp	Ref	Expression
1		pm5.32da.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) )
2	1	ex	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )
3	2	pm5.32d	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) ) )