Theorem cbvrex

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvrexw when possible. (Contributed by NM, 31-Jul-2003) (Proof shortened by Andrew Salmon, 8-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvral.1	⊢ Ⅎ 𝑦 𝜑
	cbvral.2	⊢ Ⅎ 𝑥 𝜓
	cbvral.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvrex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvral.1	⊢ Ⅎ 𝑦 𝜑
2		cbvral.2	⊢ Ⅎ 𝑥 𝜓
3		cbvral.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4		nfcv	⊢ Ⅎ 𝑥 𝐴
5		nfcv	⊢ Ⅎ 𝑦 𝐴
6	4 5 1 2 3	cbvrexf	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 𝜓 )