Theorem equsexvwOLD

Description: Obsolete version of equsexvw as of 23-Oct-2023. (Contributed by BJ, 31-May-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	equsalvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	equsexvwOLD	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		equsalvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	pm5.32i	$$⊢\left(x=y\wedge \phi \right)\leftrightarrow \left(x=y\wedge \psi \right)$$
3	2	exbii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \exists x\left(x=y\wedge \psi \right)$$
4		ax6ev	$$⊢\exists xx=y$$
5		19.41v	$$⊢\exists x\left(x=y\wedge \psi \right)\leftrightarrow \left(\exists xx=y\wedge \psi \right)$$
6	4 5	mpbiran	$$⊢\exists x\left(x=y\wedge \psi \right)\leftrightarrow \psi$$
7	3 6	bitri	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$