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Theorem trpredeq2d 25502
Description: Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
Hypothesis
Ref Expression
trpredeq2d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
trpredeq2d  |-  ( ph  -> 
TrPred ( R ,  A ,  X )  =  TrPred ( R ,  B ,  X ) )

Proof of Theorem trpredeq2d
StepHypRef Expression
1 trpredeq2d.1 . 2  |-  ( ph  ->  A  =  B )
2 trpredeq2 25499 . 2  |-  ( A  =  B  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  B ,  X
) )
31, 2syl 16 1  |-  ( ph  -> 
TrPred ( R ,  A ,  X )  =  TrPred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   TrPredctrpred 25495
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-recs 6633  df-rdg 6668  df-pred 25439  df-trpred 25496
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