Theorem sbcel1g 3987

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 3983	. 2 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
2		csbconstg 3546	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	eleq2d 2687	. 2 |- ( A e. V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> [_ A / x ]_ B e. C ) )
4	1, 3	syl5bb 272	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by:  rspcsbela  4006  csbopg  4420  fprodcllemf  14688  wunnat  16616  catcfuccl  16759  esumpfinvalf  30138  esum2dlem  30154  measiuns  30280  bj-sbel1  32900  csbfinxpg  33225  finixpnum  33394  renegclALT  34249  cdlemk35s  36225  ellimcabssub0  39849