ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difeq2d Unicode version

Theorem difeq2d 3118
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 3112 . 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 1289    \ cdif 2996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-rab 2368  df-dif 3001
This theorem is referenced by:  difeq12d  3119  phplem3  6570  phplem4  6571  phplem3g  6572  phplem4dom  6578  phplem4on  6583  fidifsnen  6586  xpfi  6640  sbthlem2  6667  sbthlemi3  6668  isbth  6676  setsvalg  11524  setsvala  11525
  Copyright terms: Public domain W3C validator