Theorem sbcel1v 3091

Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbcel1v	|- ( [. A / x ]. x e. B <-> A e. B )

Distinct variable group:   x, B

Allowed substitution hint:   A( x)

Proof of Theorem sbcel1v

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3037	. 2 |- ( [. A / x ]. x e. B -> A e. _V )
2		elex 2811	. 2 |- ( A e. B -> A e. _V )
3		dfsbcq2 3031	. . 3 |- ( y = A -> ( [ y / x ] x e. B <-> [. A / x ]. x e. B ) )
4		eleq1 2292	. . 3 |- ( y = A -> ( y e. B <-> A e. B ) )
5		clelsb1 2334	. . 3 |- ( [ y / x ] x e. B <-> y e. B )
6	3, 4, 5	vtoclbg 2862	. 2 |- ( A e. _V -> ( [. A / x ]. x e. B <-> A e. B ) )
7	1, 2, 6	pm5.21nii 709	1 |- ( [. A / x ]. x e. B <-> A e. B )

Syntax hints:   <-> wb 105   [wsb 1808   e. wcel 2200   _Vcvv 2799   [.wsbc 3028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029

This theorem is referenced by:  f1od2  6379  gropeld  15844  grstructeld2dom  15845