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

Theorem trpredeq1d 24226
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 24223 . 2  |-  ( R  =  S  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( S ,  A ,  X
) )
31, 2syl 15 1  |-  ( ph  -> 
TrPred ( R ,  A ,  X )  =  TrPred ( S ,  A ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   TrPredctrpred 24220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-recs 6388  df-rdg 6423  df-pred 24168  df-trpred 24221
  Copyright terms: Public domain W3C validator