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  5732  mpomptsx  6264  dmmpossx  6266  fmpox  6267  fmpoco  6283  xpf1o  6914  summodclem3  11564  summodclem2a  11565  summodc  11567  zsumdc  11568  fsum3  11571  sumsnf  11593  fsumcnv  11621  fisumcom2  11622  fsumshftm  11629  fisum0diag2  11631  prodmodclem3  11759  prodmodclem2a  11760  prodmodc  11762  zproddc  11763  fprodseq  11767  prodsnf  11776  fprodcnv  11809  fprodcom2fi  11810  pcmpt  12539  ctiunctlemu1st  12678  ctiunctlemu2nd  12679  ctiunctlemudc  12681  ctiunctlemfo  12683  prdsex  12973  imasex  13009  psrval  14298  fsumdvdsmul  15313