Theorem sbcel1g 3826

Description: Move proper substitution in and out of a membership relation. Note that the scope of [. A / x ]. is the wff B e. C, whereas the scope of [_ A / x ]_ is the class B. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
sbcel1g	|- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)   V( x)

Proof of Theorem sbcel1g

Step	Hyp	Ref	Expression
1		sbcel12 3822	. 2 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
2		csbconstg 3433	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	eleq2d 2524	. 2 |- ( A e. V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> [_ A / x ]_ B e. C ) )
4	1, 3	syl5bb 257	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )

Syntax hints:   -> wi 4   <-> wb 184   e. wcel 1823   [.wsbc 3324   [_csb 3420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784

This theorem is referenced by:  rspcsbela  3845  wunnat  15444  catcfuccl  15587  nbgraopALT  24626  rusgrasn  25147  esumpfinvalf  28305  esum2dlem  28321  measiuns  28425  finixpnum  30278  fprodcllemf  31830  ellimcabssub0  31862  bj-sbel1  34873  renegclALT  35091  cdlemk35s  37060