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Metamath Proof Explorer

Theorem cbvexdva

Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvexdvaw if possible. (Contributed by David Moews, 1-May-2017) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	cbvaldva.1	$⊢ (φ \land x = y) \to (ψ \leftrightarrow χ)$
	Assertion	cbvexdva	$⊢ φ \to (\exists x ψ \leftrightarrow \exists y χ)$

Proof

Step	Hyp	Ref	Expression
1		cbvaldva.1	$⊢ (φ \land x = y) \to (ψ \leftrightarrow χ)$
2		nfv	$⊢ Ⅎ y φ$
3		nfvd	$⊢ φ \to Ⅎ y ψ$
4	1	ex	$⊢ φ \to (x = y \to (ψ \leftrightarrow χ))$
5	2 3 4	cbvexd	$⊢ φ \to (\exists x ψ \leftrightarrow \exists y χ)$