Theorem sbcel1g 3787

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 3783	. 2 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
2		csbconstg 3387	. . 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 265	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )

Syntax hints:   -> wi 4   <-> wb 189   e. wcel 1897   [.wsbc 3278   [_csb 3374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-in 3422  df-ss 3429  df-nul 3743

This theorem is referenced by:  rspcsbela  3806  fprodcllemf  14060  wunnat  15909  catcfuccl  16052  nbgraopALT  25200  rusgrasn  25721  esumpfinvalf  28945  esum2dlem  28961  measiuns  29087  bj-sbel1  31551  csbopg2  31769  csbfinxpg  31824  finixpnum  31974  renegclALT  32579  cdlemk35s  34548  ellimcabssub0  37734