Theorem eqvinc

Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Proof shortened by Thierry Arnoux, 23-Jan-2022)

	Ref	Expression
Hypothesis	eqvinc.1	$$⊢A\in \mathrm{V}$$
Assertion	eqvinc	$$⊢A=B\leftrightarrow \exists x\left(x=A\wedge x=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqvinc.1	$$⊢A\in \mathrm{V}$$
2		eqvincg	$$⊢A\in \mathrm{V}\to \left(A=B\leftrightarrow \exists x\left(x=A\wedge x=B\right)\right)$$
3	1 2	ax-mp	$$⊢A=B\leftrightarrow \exists x\left(x=A\wedge x=B\right)$$