Theorem equsexv

Description: An equivalence related to implicit substitution. Version of equsex with a disjoint variable condition, which does not require ax-13 . See equsexvw for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019) Avoid ax-10 . (Revised by Gino Giotto, 18-Nov-2024)

	Ref	Expression
Hypotheses	equsalv.nf	$$⊢Ⅎx\psi$$
	equsalv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	equsexv	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		equsalv.nf	$$⊢Ⅎx\psi$$
2		equsalv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	2	biimpa	$$⊢\left(x=y\wedge \phi \right)\to \psi$$
4	1 3	exlimi	$$⊢\exists x\left(x=y\wedge \phi \right)\to \psi$$
5	1 2	equsalv	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$
6		equs4v	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$
7	5 6	sylbir	$$⊢\psi \to \exists x\left(x=y\wedge \phi \right)$$
8	4 7	impbii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$