Theorem difeq1d 3324

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 3318	. 2 |- ( A = B -> ( A \ C ) = ( B \ C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A \ C ) = ( B \ C ) )

Syntax hints:   -> wi 4   = wceq 1397   \ cdif 3197

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-dif 3202

This theorem is referenced by:  difeq12d  3326  diftpsn3  3814  phplem4  7040  phplem3g  7041  phplem4on  7053  en2other2  7406  isstruct2im  13091  isstruct2r  13092  setsfun0  13117  ptex  13346  cldval  14822  difopn  14831  cnclima  14946