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Theorem rdgeq12 6671
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 6670 . 2  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
2 rdgeq1 6669 . 2  |-  ( F  =  G  ->  rec ( F ,  B )  =  rec ( G ,  B ) )
31, 2sylan9eqr 2490 1  |-  ( ( F  =  G  /\  A  =  B )  ->  rec ( F ,  A )  =  rec ( G ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   reccrdg 6667
This theorem is referenced by:  seqomeq12  6711  seqeq3  11328  trpredeq1  25498  trpredeq2  25499  trpred0  25514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-un 3325  df-if 3740  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-iota 5418  df-fv 5462  df-recs 6633  df-rdg 6668
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