Theorem sbcel1v

Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018) Avoid ax-13 . (Revised by Wolf Lammen, 30-Apr-2023)

	Ref	Expression
Assertion	sbcel1v	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x\in B\leftrightarrow A\in B$$

Proof

Step	Hyp	Ref	Expression
1		sbcex	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x\in B\to A\in \mathrm{V}$$
2		elex	$$⊢A\in B\to A\in \mathrm{V}$$
3		dfsbcq2	$$⊢y=A\to \left(\left[y/x\right]x\in B\leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x\in B\right)$$
4		eleq1	$$⊢y=A\to \left(y\in B\leftrightarrow A\in B\right)$$
5		clelsb3	$$⊢\left[y/x\right]x\in B\leftrightarrow y\in B$$
6	3 4 5	vtoclbg	$$⊢A\in \mathrm{V}\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x\in B\leftrightarrow A\in B\right)$$
7	1 2 6	pm5.21nii	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x\in B\leftrightarrow A\in B$$