ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difeq12d Unicode version

Theorem difeq12d 3254
Description: Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
Hypotheses
Ref Expression
difeq12d.1  |-  ( ph  ->  A  =  B )
difeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
difeq12d  |-  ( ph  ->  ( A  \  C
)  =  ( B 
\  D ) )

Proof of Theorem difeq12d
StepHypRef Expression
1 difeq12d.1 . . 3  |-  ( ph  ->  A  =  B )
21difeq1d 3252 . 2  |-  ( ph  ->  ( A  \  C
)  =  ( B 
\  C ) )
3 difeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43difeq2d 3253 . 2  |-  ( ph  ->  ( B  \  C
)  =  ( B 
\  D ) )
52, 4eqtrd 2210 1  |-  ( ph  ->  ( A  \  C
)  =  ( B 
\  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    \ cdif 3126
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-dif 3131
This theorem is referenced by:  undifexmid  4190  exmidundif  4203  exmidundifim  4204
  Copyright terms: Public domain W3C validator