Theorem ifpnot23d

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

	Ref	Expression
Assertion	ifpnot23d	⊢ ( ¬ 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		notnotb	⊢ ( 𝜒 ↔ ¬ ¬ 𝜒 )
4		ifpbi23	⊢ ( ( ( 𝜓 ↔ ¬ ¬ 𝜓 ) ∧ ( 𝜒 ↔ ¬ ¬ 𝜒 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ if- ( 𝜑 , ¬ ¬ 𝜓 , ¬ ¬ 𝜒 ) ) )
5	2 3 4	mp2an	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ if- ( 𝜑 , ¬ ¬ 𝜓 , ¬ ¬ 𝜒 ) )
6	1 5	bitr4i	⊢ ( ¬ if- ( 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) ↔ if- ( 𝜑 , 𝜓 , 𝜒 ) )