Theorem csbeq1d 3147

Description: Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)

Hypothesis

Ref	Expression
csbeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
csbeq1d	|- ( ph -> [_ A / x ]_ C = [_ B / x ]_ C )

Proof of Theorem csbeq1d

Step	Hyp	Ref	Expression
1		csbeq1d.1	. 2 |- ( ph -> A = B )
2		csbeq1 3143	. 2 |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )
3	1, 2	syl 14	1 |- ( ph -> [_ A / x ]_ C = [_ B / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1398   [_csb 3140

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-sbc 3045  df-csb 3141

This theorem is referenced by:  csbidmg  3197  csbco3g  3199  fmptcof  5846  mpomptsx  6395  dmmpossx  6397  fmpox  6398  fmpoco  6414  xpf1o  7099  summodclem3  12074  summodclem2a  12075  summodc  12077  zsumdc  12078  fsum3  12081  sumsnf  12103  fsumcnv  12131  fisumcom2  12132  fsumshftm  12139  fisum0diag2  12141  prodmodclem3  12269  prodmodclem2a  12270  prodmodc  12272  zproddc  12273  fprodseq  12277  prodsnf  12286  fprodcnv  12319  fprodcom2fi  12320  pcmpt  13049  ctiunctlemu1st  13206  ctiunctlemu2nd  13207  ctiunctlemudc  13209  ctiunctlemfo  13211  prdsex  13503  imasex  13539  psrval  14863  fsumdvdsmul  15908