Theorem ifpnot23b

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

	Ref	Expression
Assertion	ifpnot23b	⊢ ( ¬ 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	⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 )
3		ifpbi2	⊢ ( ( 𝜓 ↔ ¬ ¬ 𝜓 ) → ( 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 ) )