Theorem cdeqal

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

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

Proof

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