Theorem csbeq1a 2917

Description: Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbeq1a	|- ( x = A -> B = [_ A / x ]_ B )

Proof of Theorem csbeq1a

Step	Hyp	Ref	Expression
1		csbid 2916	. 2 |- [_ x / x ]_ B = B
2		csbeq1 2912	. 2 |- ( x = A -> [_ x / x ]_ B = [_ A / x ]_ B )
3	1, 2	syl5eqr 2128	1 |- ( x = A -> B = [_ A / x ]_ B )

Syntax hints:   -> wi 4   = wceq 1285   [_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:  csbhypf  2942  csbiebt  2943  sbcnestgf  2954  cbvralcsf  2965  cbvrexcsf  2966  cbvreucsf  2967  cbvrabcsf  2968  csbing  3180  sbcbrg  3842  moop2  4014  pofun  4075  eusvnf  4211  opeliunxp  4421  elrnmpt1  4613  csbima12g  4716  fvmpts  5282  fvmpt2  5286  mptfvex  5288  fmptco  5362  fmptcof  5363  fmptcos  5364  elabrex  5429  fliftfuns  5469  csbov123g  5574  ovmpt2s  5655  csbopeq1a  5845  mpt2mptsx  5854  dmmpt2ssx  5856  fmpt2x  5857  mpt2fvex  5860  fmpt2co  5868  eqerlem  6203  qliftfuns  6256  xpf1o  6385  sumeq2d  10334  sumeq2  10335