Theorem csbeq1d 3088

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 3084	. 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 3081

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-sbc 2987  df-csb 3082

This theorem is referenced by:  csbidmg  3138  csbco3g  3140  fmptcof  5726  mpomptsx  6252  dmmpossx  6254  fmpox  6255  fmpoco  6271  xpf1o  6902  summodclem3  11526  summodclem2a  11527  summodc  11529  zsumdc  11530  fsum3  11533  sumsnf  11555  fsumcnv  11583  fisumcom2  11584  fsumshftm  11591  fisum0diag2  11593  prodmodclem3  11721  prodmodclem2a  11722  prodmodc  11724  zproddc  11725  fprodseq  11729  prodsnf  11738  fprodcnv  11771  fprodcom2fi  11772  pcmpt  12484  ctiunctlemu1st  12594  ctiunctlemu2nd  12595  ctiunctlemudc  12597  ctiunctlemfo  12599  prdsex  12883  imasex  12891  psrval  14163