Theorem exbir 45447

Description: Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 45820. (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 223	. . 3 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒))
2	1	imim2i 17	. 2 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒)))
3	2	expd 421	1 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))

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: (None)