Theorem pm5.32dav 45621

Description: Distribution of implication over biconditional (deduction form). Variant of pm5.32da 582. (Contributed by Zhi Wang, 30-Aug-2024.)

Hypothesis

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

Assertion

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

Proof of Theorem pm5.32dav

Step	Hyp	Ref	Expression
1		pm5.32dav.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
2	1	pm5.32da 582	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))
3		ancom 464	. 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
4		ancom 464	. 2 ⊢ ((𝜓 ∧ 𝜃) ↔ (𝜃 ∧ 𝜓))
5	2, 3, 4	3bitr3g 316	1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by: (None)