Theorem csbeq1 2936

Description: Analog of dfsbcq 2842 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 2842	. . 3 |- ( A = B -> ( [. A / x ]. y e. C <-> [. B / x ]. y e. C ) )
2	1	abbidv 2205	. 2 |- ( A = B -> { y | [. A / x ]. y e. C } = { y | [. B / x ]. y e. C } )
3		df-csb 2934	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
4		df-csb 2934	. 2 |- [_ B / x ]_ C = { y | [. B / x ]. y e. C }
5	2, 3, 4	3eqtr4g 2145	1 |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1289   e. wcel 1438   {cab 2074   [.wsbc 2840   [_csb 2933

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2841  df-csb 2934

This theorem is referenced by:  csbeq1d  2939  csbeq1a  2941  csbiebg  2970  sbcnestgf  2979  cbvralcsf  2990  cbvrexcsf  2991  cbvreucsf  2992  cbvrabcsf  2993  csbing  3207  disjnims  3835  sbcbrg  3892  csbopabg  3914  pofun  4137  csbima12g  4788  csbiotag  5003  fvmpts  5376  fvmpt2  5380  mptfvex  5382  fmptcof  5459  fmptcos  5460  fliftfuns  5569  csbriotag  5612  csbov123g  5679  eqerlem  6313  qliftfuns  6366  isummolem2a  10758  zisum  10761  fisum  10765  sumsnf  10790  sumsns  10796  fsum2dlemstep  10815  fisumcom2  10819  fsumshftm  10826  fisum0diag2  10828  fsumiun  10858