Theorem csbeq1d 3154

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 3150	. 2 |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )
3	1, 2	syl 14	1 |- ( ph -> [_ A / x ]_ C = [_ B / x ]_ C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   [_csb 3147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052  df-csb 3148

This theorem is used by:  csbidmg  3204  csbco3g  3206  fmptcof  5875  mpomptsx  6433  dmmpossx  6435  fmpox  6436  fmpoco  6452  xpf1o  7144  summodclem3  12166  summodclem2a  12167  summodc  12169  zsumdc  12170  fsum3  12173  sumsnf  12195  fsumcnv  12223  fisumcom2  12224  fsumshftm  12231  fisum0diag2  12233  prodmodclem3  12361  prodmodclem2a  12362  prodmodc  12364  zproddc  12365  fprodseq  12369  prodsnf  12378  fprodcnv  12411  fprodcom2fi  12412  pcmpt  13145  ctiunctlemu1st  13377  ctiunctlemu2nd  13378  ctiunctlemudc  13380  ctiunctlemfo  13382  imasex  13679  prdsex  14256  psrval  15134  fsumdvdsmul  16251