Theorem ifpbicor 40980

Description: Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020.)

Assertion

Ref	Expression
ifpbicor	⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑))

Proof of Theorem ifpbicor

Step	Hyp	Ref	Expression
1		bicom 221	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
2		ifpdfbi 1067	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
3		ifpdfbi 1067	. 2 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
4	1, 2, 3	3bitr3i 300	1 ⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 205  if-wif 1059

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 206  df-an 396  df-or 844  df-ifp 1060

This theorem is referenced by:  ifpxorcor  40981