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Theorem trpredeq1d 25440
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 25437 . 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 1649   TrPredctrpred 25434
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 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-recs 6592  df-rdg 6627  df-pred 25382  df-trpred 25435
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