Theorem difeq1d 3267

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 3261	. 2 |- ( A = B -> ( A \ C ) = ( B \ C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A \ C ) = ( B \ C ) )

Syntax hints:   -> wi 4   = wceq 1364   \ cdif 3141

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-dif 3146

This theorem is referenced by:  difeq12d  3269  diftpsn3  3748  phplem4  6882  phplem3g  6883  phplem4on  6894  en2other2  7224  isstruct2im  12521  isstruct2r  12522  setsfun0  12547  ptex  12766  cldval  14051  difopn  14060  cnclima  14175