Theorem ceqex

Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995) (Proof shortened by BJ, 1-May-2019)

	Ref	Expression
Assertion	ceqex	$$⊢x=A\to \left(\phi \leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.8a	$$⊢\left(x=A\wedge \phi \right)\to \exists x\left(x=A\wedge \phi \right)$$
2	1	ex	$$⊢x=A\to \left(\phi \to \exists x\left(x=A\wedge \phi \right)\right)$$
3		eqvisset	$$⊢x=A\to A\in \mathrm{V}$$
4		alexeqg	$$⊢A\in \mathrm{V}\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
5	3 4	syl	$$⊢x=A\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
6		sp	$$⊢\forall x\left(x=A\to \phi \right)\to \left(x=A\to \phi \right)$$
7	6	com12	$$⊢x=A\to \left(\forall x\left(x=A\to \phi \right)\to \phi \right)$$
8	5 7	sylbird	$$⊢x=A\to \left(\exists x\left(x=A\wedge \phi \right)\to \phi \right)$$
9	2 8	impbid	$$⊢x=A\to \left(\phi \leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$