Theorem csbeq1d 3014

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 3010	. 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 1332   [_csb 3007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-sbc 2914  df-csb 3008

This theorem is referenced by:  csbidmg  3061  csbco3g  3063  fmptcof  5595  mpomptsx  6103  dmmpossx  6105  fmpox  6106  fmpoco  6121  xpf1o  6746  summodclem3  11181  summodclem2a  11182  summodc  11184  zsumdc  11185  fsum3  11188  sumsnf  11210  fsumcnv  11238  fisumcom2  11239  fsumshftm  11246  fisum0diag2  11248  prodmodclem3  11376  prodmodclem2a  11377  prodmodc  11379  zproddc  11380  fprodseq  11384  ctiunctlemu1st  11983  ctiunctlemu2nd  11984  ctiunctlemudc  11986  ctiunctlemfo  11988