Theorem cdeqnot

Description: Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016)

	Ref	Expression
Hypothesis	cdeqnot.1	⊢ CondEq ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cdeqnot	⊢ CondEq ( 𝑥 = 𝑦 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		cdeqnot.1	⊢ CondEq ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2	1	cdeqri	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3	2	notbid	⊢ ( 𝑥 = 𝑦 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
4	3	cdeqi	⊢ CondEq ( 𝑥 = 𝑦 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )