Theorem impcon4bid 196

Description: A variation on impbid 183 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)

Hypotheses

Ref	Expression
impcon4bid.1	⊢ (φ → (ψ → χ))
impcon4bid.2	⊢ (φ → (¬ ψ → ¬ χ))

Assertion

Ref	Expression
impcon4bid	⊢ (φ → (ψ ↔ χ))

Proof of Theorem impcon4bid

Step	Hyp	Ref	Expression
1		impcon4bid.1	. 2 ⊢ (φ → (ψ → χ))
2		impcon4bid.2	. . 3 ⊢ (φ → (¬ ψ → ¬ χ))
3	2	con4d 97	. 2 ⊢ (φ → (χ → ψ))
4	1, 3	impbid 183	1 ⊢ (φ → (ψ ↔ χ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176

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

This proof depends on definitions:  df-bi 177

This theorem is used by:  con4bid  284