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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
StepHypRef Expression
1 difeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 difeq1 3261 . 2  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
31, 2syl 14 1  |-  ( ph  ->  ( A  \  C
)  =  ( B 
\  C ) )
Colors of variables: wff set class
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
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