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	$$⊢\psi \to \forall x\psi$$
	equsalhw.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	equsexhv	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		equsalhw.1	$$⊢\psi \to \forall x\psi$$
2		equsalhw.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	1	nf5i	$$⊢Ⅎx\psi$$
4	3 2	equsexv	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$