Theorem impbid21d 182

Description: Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)

Hypotheses

Ref	Expression
impbid21d.1	⊢ (ψ → (χ → θ))
impbid21d.2	⊢ (φ → (θ → χ))

Assertion

Ref	Expression
impbid21d	⊢ (φ → (ψ → (χ ↔ θ)))

Proof of Theorem impbid21d

Step	Hyp	Ref	Expression
1		impbid21d.1	. . 3 ⊢ (ψ → (χ → θ))
2	1	a1i 10	. 2 ⊢ (φ → (ψ → (χ → θ)))
3		impbid21d.2	. . 3 ⊢ (φ → (θ → χ))
4	3	a1d 22	. 2 ⊢ (φ → (ψ → (θ → χ)))
5	2, 4	impbidd 181	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:  impbid  183  pm5.1im  229