Theorem csbeq1d 3091

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 3087	. 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 1364   [_csb 3084

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990  df-csb 3085

This theorem is referenced by:  csbidmg  3141  csbco3g  3143  fmptcof  5729  mpomptsx  6255  dmmpossx  6257  fmpox  6258  fmpoco  6274  xpf1o  6905  summodclem3  11545  summodclem2a  11546  summodc  11548  zsumdc  11549  fsum3  11552  sumsnf  11574  fsumcnv  11602  fisumcom2  11603  fsumshftm  11610  fisum0diag2  11612  prodmodclem3  11740  prodmodclem2a  11741  prodmodc  11743  zproddc  11744  fprodseq  11748  prodsnf  11757  fprodcnv  11790  fprodcom2fi  11791  pcmpt  12512  ctiunctlemu1st  12651  ctiunctlemu2nd  12652  ctiunctlemudc  12654  ctiunctlemfo  12656  prdsex  12940  imasex  12948  psrval  14220  fsumdvdsmul  15227