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Theorem difeq2d 3189
Description: Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
Hypothesis
Ref Expression
difeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
difeq2d  |-  ( ph  ->  ( C  \  A
)  =  ( C 
\  B ) )

Proof of Theorem difeq2d
StepHypRef Expression
1 difeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 difeq2 3183 . 2  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
31, 2syl 14 1  |-  ( ph  ->  ( C  \  A
)  =  ( C 
\  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    \ cdif 3063
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-in1 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-rab 2423  df-dif 3068
This theorem is referenced by:  difeq12d  3190  phplem3  6741  phplem4  6742  phplem3g  6743  phplem4dom  6749  phplem4on  6754  fidifsnen  6757  xpfi  6811  sbthlem2  6839  sbthlemi3  6840  isbth  6848  ismkvnex  7022  setsvalg  11978  setsvala  11979  exmid1stab  13184
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