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Theorem trpredeq2d 24612
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 24609 . 2  |-  ( A  =  B  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  B ,  X
) )
31, 2syl 15 1  |-  ( ph  -> 
TrPred ( R ,  A ,  X )  =  TrPred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633   TrPredctrpred 24605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-cnv 4734  df-dm 4736  df-rn 4737  df-res 4738  df-iota 5256  df-fv 5300  df-recs 6430  df-rdg 6465  df-pred 24553  df-trpred 24606
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