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  5878  mpomptsx  6437  dmmpossx  6439  fmpox  6440  fmpoco  6456  xpf1o  7149  summodclem3  12173  summodclem2a  12174  summodc  12176  zsumdc  12177  fsum3  12180  sumsnf  12202  fsumcnv  12230  fisumcom2  12231  fsumshftm  12238  fisum0diag2  12240  prodmodclem3  12368  prodmodclem2a  12369  prodmodc  12371  zproddc  12372  fprodseq  12376  prodsnf  12385  fprodcnv  12418  fprodcom2fi  12419  pcmpt  13152  ctiunctlemu1st  13384  ctiunctlemu2nd  13385  ctiunctlemudc  13387  ctiunctlemfo  13389  imasex  13686  prdsex  14280  psrval  15162  psrass1filem  15182  fsumdvdsmul  16297