Theorem csbeq1d 3065

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 3061	. 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 1353   [_csb 3058

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2964  df-csb 3059

This theorem is referenced by:  csbidmg  3114  csbco3g  3116  fmptcof  5684  mpomptsx  6198  dmmpossx  6200  fmpox  6201  fmpoco  6217  xpf1o  6844  summodclem3  11388  summodclem2a  11389  summodc  11391  zsumdc  11392  fsum3  11395  sumsnf  11417  fsumcnv  11445  fisumcom2  11446  fsumshftm  11453  fisum0diag2  11455  prodmodclem3  11583  prodmodclem2a  11584  prodmodc  11586  zproddc  11587  fprodseq  11591  prodsnf  11600  fprodcnv  11633  fprodcom2fi  11634  pcmpt  12341  ctiunctlemu1st  12435  ctiunctlemu2nd  12436  ctiunctlemudc  12438  ctiunctlemfo  12440  prdsex  12718  imasex  12726