Theorem clelsb1f

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 ). Usage of this theorem is discouraged because it depends on ax-13 . See clelsb1fw not requiring ax-13 , but extra disjoint variables. (Contributed by Rodolfo Medina, 28-Apr-2010) (Proof shortened by Andrew Salmon, 14-Jun-2011) (Revised by Thierry Arnoux, 13-Mar-2017) (Proof shortened by Wolf Lammen, 7-May-2023) (New usage is discouraged.)

	Ref	Expression
Hypothesis	clelsb1f.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	clelsb1f	$$⊢\left[y/x\right]x\in A\leftrightarrow y\in A$$

Proof

Step	Hyp	Ref	Expression
1		clelsb1f.1	$$⊢\underset{\_}{Ⅎ}xA$$
2	1	nfcri	$$⊢Ⅎxw\in A$$
3	2	sbco2	$$⊢\left[y/x\right]\left[x/w\right]w\in A\leftrightarrow \left[y/w\right]w\in A$$
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	$$⊢\left[y/x\right]x\in A\leftrightarrow y\in A$$