Theorem csbeq1 3010

Description: Analog of dfsbcq 2915 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 2915	. . 3 |- ( A = B -> ( [. A / x ]. y e. C <-> [. B / x ]. y e. C ) )
2	1	abbidv 2258	. 2 |- ( A = B -> { y | [. A / x ]. y e. C } = { y | [. B / x ]. y e. C } )
3		df-csb 3008	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
4		df-csb 3008	. 2 |- [_ B / x ]_ C = { y | [. B / x ]. y e. C }
5	2, 3, 4	3eqtr4g 2198	1 |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   {cab 2126   [.wsbc 2913   [_csb 3007

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-sbc 2914  df-csb 3008

This theorem is referenced by:  csbeq1d  3014  csbeq1a  3016  csbiebg  3047  sbcnestgf  3056  cbvralcsf  3067  cbvrexcsf  3068  cbvreucsf  3069  cbvrabcsf  3070  csbing  3288  disjnims  3929  sbcbrg  3990  csbopabg  4014  pofun  4242  csbima12g  4908  csbiotag  5124  fvmpts  5507  fvmpt2  5512  mptfvex  5514  elfvmptrab1  5523  fmptcof  5595  fmptcos  5596  fliftfuns  5707  csbriotag  5750  csbov123g  5817  eqerlem  6468  qliftfuns  6521  summodclem2a  11182  zsumdc  11185  fsum3  11188  sumsnf  11210  sumsns  11216  fsum2dlemstep  11235  fisumcom2  11239  fsumshftm  11246  fisum0diag2  11248  fsumiun  11278  ctiunctlemf  11987  mulcncflem  12798