Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trpredeq3d Structured version   Unicode version

Theorem trpredeq3d 25505
Description: Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
Hypothesis
Ref Expression
trpredeq3d.1  |-  ( ph  ->  X  =  Y )
Assertion
Ref Expression
trpredeq3d  |-  ( ph  -> 
TrPred ( R ,  A ,  X )  =  TrPred ( R ,  A ,  Y ) )

Proof of Theorem trpredeq3d
StepHypRef Expression
1 trpredeq3d.1 . 2  |-  ( ph  ->  X  =  Y )
2 trpredeq3 25502 . 2  |-  ( X  =  Y  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  A ,  Y
) )
31, 2syl 16 1  |-  ( ph  -> 
TrPred ( R ,  A ,  X )  =  TrPred ( R ,  A ,  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   TrPredctrpred 25497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fv 5464  df-recs 6635  df-rdg 6670  df-pred 25441  df-trpred 25498
  Copyright terms: Public domain W3C validator