Metamath Proof Explorer


Theorem ifpnorcor

Description: Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020)

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

Proof

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