Theorem pm5.32dar 49904

Description: Reverse distribution of implication over biconditional (deduction form). (Contributed by Zhi Wang, 6-Sep-2024.)

Hypothesis

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

Assertion

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

Proof of Theorem pm5.32dar

Step	Hyp	Ref	Expression
1		pm5.32dar.1	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))
2		pm5.32 584	. . 3 ⊢ ((𝜓 → (𝜒 ↔ 𝜃)) ↔ ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))
3	1, 2	sylibr 237	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
4	3	imp 412	1 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  clddisj  50011