Theorem impbid21d 128

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 9	. 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 127	1 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  impbid  129  pm5.1im  173