Theorem difeq1d 3280

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 3274	. 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 3154

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-dif 3159

This theorem is referenced by:  difeq12d  3282  diftpsn3  3763  phplem4  6916  phplem3g  6917  phplem4on  6928  en2other2  7263  isstruct2im  12688  isstruct2r  12689  setsfun0  12714  ptex  12935  cldval  14335  difopn  14344  cnclima  14459