Theorem csbeq1d 3064

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 3060	. 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 3057

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 2963  df-csb 3058

This theorem is referenced by:  csbidmg  3113  csbco3g  3115  fmptcof  5683  mpomptsx  6197  dmmpossx  6199  fmpox  6200  fmpoco  6216  xpf1o  6843  summodclem3  11387  summodclem2a  11388  summodc  11390  zsumdc  11391  fsum3  11394  sumsnf  11416  fsumcnv  11444  fisumcom2  11445  fsumshftm  11452  fisum0diag2  11454  prodmodclem3  11582  prodmodclem2a  11583  prodmodc  11585  zproddc  11586  fprodseq  11590  prodsnf  11599  fprodcnv  11632  fprodcom2fi  11633  pcmpt  12340  ctiunctlemu1st  12434  ctiunctlemu2nd  12435  ctiunctlemudc  12437  ctiunctlemfo  12439  prdsex  12717  imasex  12725