Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trpredeq1d Unicode version

Theorem trpredeq1d 25251
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 25248 . 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 25245
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
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 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-cnv 4827  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fv 5403  df-recs 6570  df-rdg 6605  df-pred 25193  df-trpred 25246
  Copyright terms: Public domain W3C validator