Theorem pm5.32rd 447

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

Hypothesis

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

Assertion

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

Proof of Theorem pm5.32rd

Step	Hyp	Ref	Expression
1		pm5.32d.1	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
2	1	pm5.32d 446	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))
3		ancom 264	. 2 ⊢ ((𝜒 ∧ 𝜓) ↔ (𝜓 ∧ 𝜒))
4		ancom 264	. 2 ⊢ ((𝜃 ∧ 𝜓) ↔ (𝜓 ∧ 𝜃))
5	2, 3, 4	3bitr4g 222	1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  anbi1d  461  pm5.71dc  951  1idprl  7531  1idpru  7532