Theorem cbvrmov

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Alexander van der Vekens, 17-Jun-2017) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cbvralv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvrmov	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑦 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvralv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑦 𝜑
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	2 3 1	cbvrmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑦 ∈ 𝐴 𝜓 )