Theorem pm5.32rd

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 25-Dec-2004)

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

Proof

Step	Hyp	Ref	Expression
1		pm5.32d.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )
2	1	pm5.32d	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) ) )
3		ancom	⊢ ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜒 ) )
4		ancom	⊢ ( ( 𝜃 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜃 ) )
5	2 3 4	3bitr4g	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜃 ∧ 𝜓 ) ) )