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  5730  mpomptsx  6256  dmmpossx  6258  fmpox  6259  fmpoco  6275  xpf1o  6906  summodclem3  11547  summodclem2a  11548  summodc  11550  zsumdc  11551  fsum3  11554  sumsnf  11576  fsumcnv  11604  fisumcom2  11605  fsumshftm  11612  fisum0diag2  11614  prodmodclem3  11742  prodmodclem2a  11743  prodmodc  11745  zproddc  11746  fprodseq  11750  prodsnf  11759  fprodcnv  11792  fprodcom2fi  11793  pcmpt  12522  ctiunctlemu1st  12661  ctiunctlemu2nd  12662  ctiunctlemudc  12664  ctiunctlemfo  12666  prdsex  12950  imasex  12958  psrval  14230  fsumdvdsmul  15237