Theorem difeq1d 3245

Description: Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Hypothesis

Ref	Expression
difeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
difeq1d	|- ( ph -> ( A \ C ) = ( B \ C ) )

Proof of Theorem difeq1d

Step	Hyp	Ref	Expression
1		difeq1d.1	. 2 |- ( ph -> A = B )
2		difeq1 3239	. 2 |- ( A = B -> ( A \ C ) = ( B \ C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A \ C ) = ( B \ C ) )

Syntax hints:   -> wi 4   = wceq 1349   \ cdif 3119

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-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-rab 2458  df-dif 3124

This theorem is referenced by:  difeq12d  3247  diftpsn3  3722  phplem4  6837  phplem3g  6838  phplem4on  6849  en2other2  7177  isstruct2im  12430  isstruct2r  12431  setsfun0  12456  cldval  12978  difopn  12987  cnclima  13102