Theorem difeq1d 3193

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 3187	. 2 |- ( A = B -> ( A \ C ) = ( B \ C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A \ C ) = ( B \ C ) )

Syntax hints:   -> wi 4   = wceq 1331   \ cdif 3068

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-dif 3073

This theorem is referenced by:  difeq12d  3195  diftpsn3  3661  phplem4  6749  phplem3g  6750  phplem4on  6761  en2other2  7052  isstruct2im  11969  isstruct2r  11970  setsfun0  11995  cldval  12268  difopn  12277  cnclima  12392