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Theorem predeq2 24728
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 3439 . 2  |-  ( A  =  B  ->  ( A  i^i  ( `' R " { X } ) )  =  ( B  i^i  ( `' R " { X } ) ) )
2 df-pred 24726 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
3 df-pred 24726 . 2  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
41, 2, 33eqtr4g 2415 1  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    i^i cin 3227   {csn 3716   `'ccnv 4770   "cima 4774   Predcpred 24725
This theorem is referenced by:  prednn  24759  prednn0  24760  trpredeq2  24782  frmin  24800  wfrlem5  24818  frrlem5  24843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-in 3235  df-pred 24726
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