Theorem exbir 1424

Description: Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.)

Assertion

Ref	Expression
exbir	⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))

Proof of Theorem exbir

Step	Hyp	Ref	Expression
1		biimpr 129	. . 3 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒))
2	1	imim2i 12	. 2 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒)))
3	2	expd 256	1 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))

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: (None)