Theorem sbc5

Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993) (Revised by Mario Carneiro, 12-Oct-2016)

	Ref	Expression
Assertion	sbc5	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\left(x=A\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcex	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to A\in \mathrm{V}$$
2		exsimpl	$$⊢\exists x\left(x=A\wedge \phi \right)\to \exists xx=A$$
3		isset	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$
4	2 3	sylibr	$$⊢\exists x\left(x=A\wedge \phi \right)\to A\in \mathrm{V}$$
5		dfsbcq2	$$⊢y=A\to \left(\left[y/x\right]\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
6		eqeq2	$$⊢y=A\to \left(x=y\leftrightarrow x=A\right)$$
7	6	anbi1d	$$⊢y=A\to \left(\left(x=y\wedge \phi \right)\leftrightarrow \left(x=A\wedge \phi \right)\right)$$
8	7	exbidv	$$⊢y=A\to \left(\exists x\left(x=y\wedge \phi \right)\leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
9		sb5	$$⊢\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)$$
10	5 8 9	vtoclbg	$$⊢A\in \mathrm{V}\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
11	1 4 10	pm5.21nii	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\left(x=A\wedge \phi \right)$$