Theorem ceqsexvOLD

Description: Obsolete version of ceqsexv as of 12-Oct-2024. (Contributed by NM, 2-Mar-1995) Avoid ax-12 . (Revised by Gino Giotto, 12-Oct-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ceqsexvOLD.1	$$⊢A\in \mathrm{V}$$
	ceqsexvOLD.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	ceqsexvOLD	$$⊢\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		ceqsexvOLD.1	$$⊢A\in \mathrm{V}$$
2		ceqsexvOLD.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3	2	biimpa	$$⊢\left(x=A\wedge \phi \right)\to \psi$$
4	3	exlimiv	$$⊢\exists x\left(x=A\wedge \phi \right)\to \psi$$
5	2	biimprcd	$$⊢\psi \to \left(x=A\to \phi \right)$$
6	5	alrimiv	$$⊢\psi \to \forall x\left(x=A\to \phi \right)$$
7	1	isseti	$$⊢\exists xx=A$$
8		exintr	$$⊢\forall x\left(x=A\to \phi \right)\to \left(\exists xx=A\to \exists x\left(x=A\wedge \phi \right)\right)$$
9	6 7 8	mpisyl	$$⊢\psi \to \exists x\left(x=A\wedge \phi \right)$$
10	4 9	impbii	$$⊢\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi$$