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Metamath Proof Explorer

Theorem equsexhv

Description: An equivalence related to implicit substitution. Version of equsexh with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	equsalhw.1	$⊢ ψ \to \forall x ψ$
	equsalhw.2	$⊢ x = y \to (φ \leftrightarrow ψ)$
Assertion	equsexhv	$⊢ \exists x (x = y \land φ) \leftrightarrow ψ$

Proof

Step	Hyp	Ref	Expression
1		equsalhw.1	$⊢ ψ \to \forall x ψ$
2		equsalhw.2	$⊢ x = y \to (φ \leftrightarrow ψ)$
3	1	nf5i	$⊢ Ⅎ x ψ$
4	3 2	equsexv	$⊢ \exists x (x = y \land φ) \leftrightarrow ψ$