Theorem ifpxorcor 40709

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

Assertion

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

Proof of Theorem ifpxorcor

Step	Hyp	Ref	Expression
1		ifpbicor 40708	. 2 ⊢ (if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜓) ↔ if-(¬ 𝜓, 𝜑, ¬ 𝜑))
2		notnotb 318	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
3		ifpbi3 40701	. . 3 ⊢ ((𝜓 ↔ ¬ ¬ 𝜓) → (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜓)))
4	2, 3	ax-mp 5	. 2 ⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜓))
5		ifpn 1074	. 2 ⊢ (if-(𝜓, ¬ 𝜑, 𝜑) ↔ if-(¬ 𝜓, 𝜑, ¬ 𝜑))
6	1, 4, 5	3bitr4i 306	1 ⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 209  if-wif 1063

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 210  df-an 400  df-or 848  df-ifp 1064

This theorem is referenced by: (None)