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Theorem preddif 27788
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 3706 . 2  |-  ( ( A  \  B )  i^i  ( `' R " { X } ) )  =  ( ( A  i^i  ( `' R " { X } ) )  \ 
( B  i^i  ( `' R " { X } ) ) )
2 df-pred 27761 . 2  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( ( A  \  B )  i^i  ( `' R " { X } ) )
3 df-pred 27761 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
4 df-pred 27761 . . 3  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
53, 4difeq12i 3572 . 2  |-  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )  =  ( ( A  i^i  ( `' R " { X } ) )  \ 
( B  i^i  ( `' R " { X } ) ) )
61, 2, 53eqtr4i 2490 1  |-  Pred ( R ,  ( A  \  B ) ,  X
)  =  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  B ,  X ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    \ cdif 3425    i^i cin 3427   {csn 3977   `'ccnv 4939   "cima 4943   Predcpred 27760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rab 2804  df-v 3072  df-dif 3431  df-in 3435  df-pred 27761
This theorem is referenced by:  wfrlem8  27867
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