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

Theorem rdgeq12 6971
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 6970 . 2  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
2 rdgeq1 6969 . 2  |-  ( F  =  G  ->  rec ( F ,  B )  =  rec ( G ,  B ) )
31, 2sylan9eqr 2514 1  |-  ( ( F  =  G  /\  A  =  B )  ->  rec ( F ,  A )  =  rec ( G ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   reccrdg 6967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-un 3433  df-if 3892  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-iota 5481  df-fv 5526  df-recs 6934  df-rdg 6968
This theorem is referenced by:  seqomeq12  7011  seqeq3  11914  trpredeq1  27820  trpredeq2  27821  trpred0  27836
  Copyright terms: Public domain W3C validator