Theorem sbcel1v 2876

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 2823	. 2 |- ( [. A / x ]. x e. B -> A e. _V )
2		elex 2610	. 2 |- ( A e. B -> A e. _V )
3		dfsbcq2 2818	. . 3 |- ( y = A -> ( [ y / x ] x e. B <-> [. A / x ]. x e. B ) )
4		eleq1 2141	. . 3 |- ( y = A -> ( y e. B <-> A e. B ) )
5		clelsb3 2183	. . 3 |- ( [ y / x ] x e. B <-> y e. B )
6	3, 4, 5	vtoclbg 2659	. 2 |- ( A e. _V -> ( [. A / x ]. x e. B <-> A e. B ) )
7	1, 2, 6	pm5.21nii 652	1 |- ( [. A / x ]. x e. B <-> A e. B )

Syntax hints:   <-> wb 103   e. wcel 1433   [wsb 1685   _Vcvv 2601   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  f1od2  5876