Theorem impbid21d 214

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	. 2 ⊢ (𝜓 → (𝜒 → 𝜃))
2		impbid21d.2	. 2 ⊢ (𝜑 → (𝜃 → 𝜒))
3		impbi 211	. 2 ⊢ ((𝜒 → 𝜃) → ((𝜃 → 𝜒) → (𝜒 ↔ 𝜃)))
4	1, 2, 3	syl2imc 42	1 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

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

This theorem is used by:  impbid  215  pm5.1im  266  anbiim  652  nanass  1540  mobi  2575