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Metamath Proof Explorer

Theorem cbvrexv

Description: Change the bound variable of a restricted existential quantifier using implicit substitution. See cbvrexvw based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvrexvw when possible. (Contributed by NM, 2-Jun-1998) (New usage is discouraged.)

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