Theorem csbeq1 2912

Description: Analog of dfsbcq 2818 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 2818	. . 3 |- ( A = B -> ( [. A / x ]. y e. C <-> [. B / x ]. y e. C ) )
2	1	abbidv 2197	. 2 |- ( A = B -> { y | [. A / x ]. y e. C } = { y | [. B / x ]. y e. C } )
3		df-csb 2910	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
4		df-csb 2910	. 2 |- [_ B / x ]_ C = { y | [. B / x ]. y e. C }
5	2, 3, 4	3eqtr4g 2139	1 |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434   {cab 2068   [.wsbc 2816   [_csb 2909

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-sbc 2817  df-csb 2910

This theorem is referenced by:  csbeq1d  2915  csbeq1a  2917  csbiebg  2946  sbcnestgf  2954  cbvralcsf  2965  cbvrexcsf  2966  cbvreucsf  2967  cbvrabcsf  2968  csbing  3180  sbcbrg  3842  csbopabg  3864  pofun  4075  csbima12g  4716  csbiotag  4925  fvmpts  5282  fvmpt2  5286  mptfvex  5288  fmptcof  5363  fmptcos  5364  fliftfuns  5469  csbriotag  5511  csbov123g  5574  eqerlem  6203  qliftfuns  6256