Metamath Proof Explorer


Theorem ifpnot23c

Description: Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020)

Ref Expression
Assertion ifpnot23c ( ¬ if- ( 𝜑 , 𝜓 , ¬ 𝜒 ) ↔ if- ( 𝜑 , ¬ 𝜓 , 𝜒 ) )

Proof

Step Hyp Ref Expression
1 ifpnot23 ( ¬ if- ( 𝜑 , 𝜓 , ¬ 𝜒 ) ↔ if- ( 𝜑 , ¬ 𝜓 , ¬ ¬ 𝜒 ) )
2 notnotb ( 𝜒 ↔ ¬ ¬ 𝜒 )
3 ifpbi3 ( ( 𝜒 ↔ ¬ ¬ 𝜒 ) → ( if- ( 𝜑 , ¬ 𝜓 , 𝜒 ) ↔ if- ( 𝜑 , ¬ 𝜓 , ¬ ¬ 𝜒 ) ) )
4 2 3 ax-mp ( if- ( 𝜑 , ¬ 𝜓 , 𝜒 ) ↔ if- ( 𝜑 , ¬ 𝜓 , ¬ ¬ 𝜒 ) )
5 1 4 bitr4i ( ¬ if- ( 𝜑 , 𝜓 , ¬ 𝜒 ) ↔ if- ( 𝜑 , ¬ 𝜓 , 𝜒 ) )