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Theorem trpredeq1d 25493
Description: Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
Hypothesis
Ref Expression
trpredeq1d.1  |-  ( ph  ->  R  =  S )
Assertion
Ref Expression
trpredeq1d  |-  ( ph  -> 
TrPred ( R ,  A ,  X )  =  TrPred ( S ,  A ,  X ) )

Proof of Theorem trpredeq1d
StepHypRef Expression
1 trpredeq1d.1 . 2  |-  ( ph  ->  R  =  S )
2 trpredeq1 25490 . 2  |-  ( R  =  S  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( S ,  A ,  X
) )
31, 2syl 16 1  |-  ( ph  -> 
TrPred ( R ,  A ,  X )  =  TrPred ( S ,  A ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   TrPredctrpred 25487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-recs 6625  df-rdg 6660  df-pred 25431  df-trpred 25488
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