Theorem sbcel1v 2971

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 2917	. 2 |- ( [. A / x ]. x e. B -> A e. _V )
2		elex 2697	. 2 |- ( A e. B -> A e. _V )
3		dfsbcq2 2912	. . 3 |- ( y = A -> ( [ y / x ] x e. B <-> [. A / x ]. x e. B ) )
4		eleq1 2202	. . 3 |- ( y = A -> ( y e. B <-> A e. B ) )
5		clelsb3 2244	. . 3 |- ( [ y / x ] x e. B <-> y e. B )
6	3, 4, 5	vtoclbg 2747	. 2 |- ( A e. _V -> ( [. A / x ]. x e. B <-> A e. B ) )
7	1, 2, 6	pm5.21nii 693	1 |- ( [. A / x ]. x e. B <-> A e. B )

Syntax hints:   <-> wb 104   e. wcel 1480   [wsb 1735   _Vcvv 2686   [.wsbc 2909

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910

This theorem is referenced by:  f1od2  6132