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Theorem predeq2 25388
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predeq2  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )

Proof of Theorem predeq2
StepHypRef Expression
1 ineq1 3499 . 2  |-  ( A  =  B  ->  ( A  i^i  ( `' R " { X } ) )  =  ( B  i^i  ( `' R " { X } ) ) )
2 df-pred 25386 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
3 df-pred 25386 . 2  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
41, 2, 33eqtr4g 2465 1  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    i^i cin 3283   {csn 3778   `'ccnv 4840   "cima 4844   Predcpred 25385
This theorem is referenced by:  prednn  25419  prednn0  25420  trpredeq2  25442  frmin  25460  wfrlem5  25478  frrlem5  25503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-in 3291  df-pred 25386
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