Theorem clelsb1fw

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 ). Version of clelsb1f with a disjoint variable condition, which does not require ax-13 . (Contributed by Rodolfo Medina, 28-Apr-2010) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypothesis	clelsb1fw.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	clelsb1fw	$$⊢\left[y/x\right]x\in A\leftrightarrow y\in A$$

Proof

Step	Hyp	Ref	Expression
1		clelsb1fw.1	$$⊢\underset{\_}{Ⅎ}xA$$
2	1	nfcri	$$⊢Ⅎxw\in A$$
3	2	sbco2v	$$⊢\left[y/x\right]\left[x/w\right]w\in A\leftrightarrow \left[y/w\right]w\in A$$
4		clelsb1	$$⊢\left[x/w\right]w\in A\leftrightarrow x\in A$$
5	4	sbbii	$$⊢\left[y/x\right]\left[x/w\right]w\in A\leftrightarrow \left[y/x\right]x\in A$$
6		clelsb1	$$⊢\left[y/w\right]w\in A\leftrightarrow y\in A$$
7	3 5 6	3bitr3i	$$⊢\left[y/x\right]x\in A\leftrightarrow y\in A$$