Theorem csbeq1 3006

Description: Analog of dfsbcq 2911 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 2911	. . 3 |- ( A = B -> ( [. A / x ]. y e. C <-> [. B / x ]. y e. C ) )
2	1	abbidv 2257	. 2 |- ( A = B -> { y | [. A / x ]. y e. C } = { y | [. B / x ]. y e. C } )
3		df-csb 3004	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
4		df-csb 3004	. 2 |- [_ B / x ]_ C = { y | [. B / x ]. y e. C }
5	2, 3, 4	3eqtr4g 2197	1 |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   {cab 2125   [.wsbc 2909   [_csb 3003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-sbc 2910  df-csb 3004

This theorem is referenced by:  csbeq1d  3010  csbeq1a  3012  csbiebg  3042  sbcnestgf  3051  cbvralcsf  3062  cbvrexcsf  3063  cbvreucsf  3064  cbvrabcsf  3065  csbing  3283  disjnims  3921  sbcbrg  3982  csbopabg  4006  pofun  4234  csbima12g  4900  csbiotag  5116  fvmpts  5499  fvmpt2  5504  mptfvex  5506  elfvmptrab1  5515  fmptcof  5587  fmptcos  5588  fliftfuns  5699  csbriotag  5742  csbov123g  5809  eqerlem  6460  qliftfuns  6513  summodclem2a  11150  zsumdc  11153  fsum3  11156  sumsnf  11178  sumsns  11184  fsum2dlemstep  11203  fisumcom2  11207  fsumshftm  11214  fisum0diag2  11216  fsumiun  11246  ctiunctlemf  11951  mulcncflem  12759