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Theorem preddif 29511
Description: Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
preddif  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )

Proof of Theorem preddif
StepHypRef Expression
1 indifdir 3751 . 2  |-  ( ( A  \  B )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  \ 
( B  i^i  ( `' R " { X } ) ) )
2 df-pred 29484 . 2  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( ( A  \  B )  i^i  ( `' R " { X } ) )
3 df-pred 29484 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
4 df-pred 29484 . . 3  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
53, 4difeq12i 3606 . 2  |-  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )  =  ( ( A  i^i  ( `' R " { X } ) )  \ 
( B  i^i  ( `' R " { X } ) ) )
61, 2, 53eqtr4i 2493 1  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    \ cdif 3458    i^i cin 3460   {csn 4016   `'ccnv 4987   "cima 4991   Predcpred 29483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-in 3468  df-pred 29484
This theorem is referenced by:  wfrlem8  29590
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