Theorem csbeq1 2911

Description: Analog of dfsbcq 2817 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbeq1	|- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )

Proof of Theorem csbeq1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 2817	. . 3 |- ( A = B -> ( [. A / x ]. y e. C <-> [. B / x ]. y e. C ) )
2	1	abbidv 2196	. 2 |- ( A = B -> { y | [. A / x ]. y e. C } = { y | [. B / x ]. y e. C } )
3		df-csb 2909	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
4		df-csb 2909	. 2 |- [_ B / x ]_ C = { y | [. B / x ]. y e. C }
5	2, 3, 4	3eqtr4g 2138	1 |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {cab 2067   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-sbc 2816  df-csb 2909

This theorem is referenced by:  csbeq1d  2914  csbeq1a  2916  csbiebg  2945  sbcnestgf  2953  cbvralcsf  2964  cbvrexcsf  2965  cbvreucsf  2966  cbvrabcsf  2967  csbing  3173  sbcbrg  3834  csbopabg  3856  pofun  4067  csbima12g  4706  csbiotag  4915  fvmpts  5271  fvmpt2  5275  mptfvex  5277  fmptcof  5352  fmptcos  5353  fliftfuns  5458  csbriotag  5500  csbov123g  5563  eqerlem  6160  qliftfuns  6213