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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	$⊢ x = y \to (φ \leftrightarrow ψ)$
	Assertion	cbvrexv	$⊢ \exists x \in A φ \leftrightarrow \exists y \in A ψ$

Proof

Step	Hyp	Ref	Expression
1		cbvralv.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
2		nfv	$⊢ Ⅎ y φ$
3		nfv	$⊢ Ⅎ x ψ$
4	2 3 1	cbvrex	$⊢ \exists x \in A φ \leftrightarrow \exists y \in A ψ$