Theorem ifpnancor 41088

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

Assertion

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

Proof of Theorem ifpnancor

Step	Hyp	Ref	Expression
1		ifpancor 41071	. . 3 ⊢ (if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))
2	1	notbii 320	. 2 ⊢ (¬ if-(𝜑, 𝜓, 𝜑) ↔ ¬ if-(𝜓, 𝜑, 𝜓))
3		ifpnot23 41085	. 2 ⊢ (¬ if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜑))
4		ifpnot23 41085	. 2 ⊢ (¬ if-(𝜓, 𝜑, 𝜓) ↔ if-(𝜓, ¬ 𝜑, ¬ 𝜓))
5	2, 3, 4	3bitr3i 301	1 ⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜑, ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 205  if-wif 1060

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 397  df-or 845  df-ifp 1061

This theorem is referenced by: (None)