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  7150  summodclem3  12171  summodclem2a  12172  summodc  12174  zsumdc  12175  fsum3  12178  sumsnf  12200  fsumcnv  12228  fisumcom2  12229  fsumshftm  12236  fisum0diag2  12238  prodmodclem3  12366  prodmodclem2a  12367  prodmodc  12369  zproddc  12370  fprodseq  12374  prodsnf  12383  fprodcnv  12416  fprodcom2fi  12417  pcmpt  13150  ctiunctlemu1st  13382  ctiunctlemu2nd  13383  ctiunctlemudc  13385  ctiunctlemfo  13387  imasex  13684  prdsex  14278  psrval  15160  psrass1filem  15186  fsumdvdsmul  16302