MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rdgeq12 Unicode version

Theorem rdgeq12 6442
Description: Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.)
Assertion
Ref Expression
rdgeq12  |-  ( ( F  =  G  /\  A  =  B )  ->  rec ( F ,  A )  =  rec ( G ,  B ) )

Proof of Theorem rdgeq12
StepHypRef Expression
1 rdgeq2 6441 . 2  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
2 rdgeq1 6440 . 2  |-  ( F  =  G  ->  rec ( F ,  B )  =  rec ( G ,  B ) )
31, 2sylan9eqr 2350 1  |-  ( ( F  =  G  /\  A  =  B )  ->  rec ( F ,  A )  =  rec ( G ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632   reccrdg 6438
This theorem is referenced by:  seqomeq12  6482  seqeq3  11067  trpredeq1  24294  trpredeq2  24295  trpred0  24310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-iota 5235  df-fv 5279  df-recs 6404  df-rdg 6439
  Copyright terms: Public domain W3C validator