Theorem sbcel1v 3052

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 2998	. 2 |- ( [. A / x ]. x e. B -> A e. _V )
2		elex 2774	. 2 |- ( A e. B -> A e. _V )
3		dfsbcq2 2992	. . 3 |- ( y = A -> ( [ y / x ] x e. B <-> [. A / x ]. x e. B ) )
4		eleq1 2259	. . 3 |- ( y = A -> ( y e. B <-> A e. B ) )
5		clelsb1 2301	. . 3 |- ( [ y / x ] x e. B <-> y e. B )
6	3, 4, 5	vtoclbg 2825	. 2 |- ( A e. _V -> ( [. A / x ]. x e. B <-> A e. B ) )
7	1, 2, 6	pm5.21nii 705	1 |- ( [. A / x ]. x e. B <-> A e. B )

Syntax hints:   <-> wb 105   [wsb 1776   e. wcel 2167   _Vcvv 2763   [.wsbc 2989

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  f1od2  6293