Theorem impbidd 181

Description: Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Hypotheses

Ref	Expression
impbidd.1	⊢ (φ → (ψ → (χ → θ)))
impbidd.2	⊢ (φ → (ψ → (θ → χ)))

Assertion

Ref	Expression
impbidd	⊢ (φ → (ψ → (χ ↔ θ)))

Proof of Theorem impbidd

Step	Hyp	Ref	Expression
1		impbidd.1	. 2 ⊢ (φ → (ψ → (χ → θ)))
2		impbidd.2	. 2 ⊢ (φ → (ψ → (θ → χ)))
3		bi3 179	. 2 ⊢ ((χ → θ) → ((θ → χ) → (χ ↔ θ)))
4	1, 2, 3	syl6c 60	1 ⊢ (φ → (ψ → (χ ↔ θ)))

Syntax hints:   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  impbid21d  182  pm5.74  235