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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
StepHypRef Expression
1 difeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 difeq1 3187 . 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 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  7057  isstruct2im  11983  isstruct2r  11984  setsfun0  12009  cldval  12282  difopn  12291  cnclima  12406
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