Theorem pm5.32d

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996)

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

Proof

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