Theorem cdeqal1

Description: Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 11-Aug-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cdeqnot.1	⊢ CondEq ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cdeqal1	⊢ CondEq ( 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) )

Proof

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