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)

	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	pm5.32i	$$⊢\left(x=y\wedge \phi \right)\leftrightarrow \left(x=y\wedge \psi \right)$$
4	3	exbii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \exists x\left(x=y\wedge \psi \right)$$
5		ax6ev	$$⊢\exists xx=y$$
6	1	19.41	$$⊢\exists x\left(x=y\wedge \psi \right)\leftrightarrow \left(\exists xx=y\wedge \psi \right)$$
7	5 6	mpbiran	$$⊢\exists x\left(x=y\wedge \psi \right)\leftrightarrow \psi$$
8	4 7	bitri	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$