Theorem pm5.74rd 183

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 19-Mar-1997.)

Hypothesis

Ref	Expression
pm5.74rd.1	⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))

Assertion

Ref	Expression
pm5.74rd	⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))

Proof of Theorem pm5.74rd

Step	Hyp	Ref	Expression
1		pm5.74rd.1	. 2 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
2		pm5.74 179	. 2 ⊢ ((𝜓 → (𝜒 ↔ 𝜃)) ↔ ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
3	1, 2	sylibr 134	1 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  imbibi  252  pm5.35  918