Theorem csbeq1 3127

Description: Analog of dfsbcq 3030 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 3030	. . 3 |- ( A = B -> ( [. A / x ]. y e. C <-> [. B / x ]. y e. C ) )
2	1	abbidv 2347	. 2 |- ( A = B -> { y | [. A / x ]. y e. C } = { y | [. B / x ]. y e. C } )
3		df-csb 3125	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
4		df-csb 3125	. 2 |- [_ B / x ]_ C = { y | [. B / x ]. y e. C }
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {cab 2215   [.wsbc 3028   [_csb 3124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029  df-csb 3125

This theorem is referenced by:  csbeq1d  3131  csbeq1a  3133  csbiebg  3167  sbcnestgf  3176  cbvralcsf  3187  cbvrexcsf  3188  cbvreucsf  3189  cbvrabcsf  3190  csbing  3411  disjnims  4074  sbcbrg  4138  csbopabg  4162  pofun  4404  csbima12g  5092  csbiotag  5314  fvmpts  5717  fvmpt2  5723  mptfvex  5725  elfvmptrab1  5734  fmptcof  5807  fmptcos  5808  fliftfuns  5931  csbriotag  5977  riotaeqimp  5988  csbov123g  6049  elovmporab1w  6215  eqerlem  6724  qliftfuns  6779  summodclem2a  11913  zsumdc  11916  fsum3  11919  sumsnf  11941  sumsns  11947  fsum2dlemstep  11966  fisumcom2  11970  fsumshftm  11977  fisum0diag2  11979  fsumiun  12009  prodsnf  12124  fprodm1s  12133  fprodp1s  12134  prodsns  12135  fprod2dlemstep  12154  fprodcom2fi  12158  pcmptdvds  12889  ctiunctlemf  13030  mulcncflem  15302  fsumdvdsmul  15686