Theorem ifpnancor

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

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

Proof

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