Theorem sbcel1v 3013

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 2959	. 2 |- ( [. A / x ]. x e. B -> A e. _V )
2		elex 2737	. 2 |- ( A e. B -> A e. _V )
3		dfsbcq2 2954	. . 3 |- ( y = A -> ( [ y / x ] x e. B <-> [. A / x ]. x e. B ) )
4		eleq1 2229	. . 3 |- ( y = A -> ( y e. B <-> A e. B ) )
5		clelsb1 2271	. . 3 |- ( [ y / x ] x e. B <-> y e. B )
6	3, 4, 5	vtoclbg 2787	. 2 |- ( A e. _V -> ( [. A / x ]. x e. B <-> A e. B ) )
7	1, 2, 6	pm5.21nii 694	1 |- ( [. A / x ]. x e. B <-> A e. B )

Syntax hints:   <-> wb 104   [wsb 1750   e. wcel 2136   _Vcvv 2726   [.wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by:  f1od2  6203