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Theorem rdgeq12 7015
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 7014 . 2  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
2 rdgeq1 7013 . 2  |-  ( F  =  G  ->  rec ( F ,  B )  =  rec ( G ,  B ) )
31, 2sylan9eqr 2455 1  |-  ( ( F  =  G  /\  A  =  B )  ->  rec ( F ,  A )  =  rec ( G ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399   reccrdg 7011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-un 3407  df-if 3871  df-uni 4177  df-br 4381  df-opab 4439  df-mpt 4440  df-iota 5473  df-fv 5517  df-recs 6978  df-rdg 7012
This theorem is referenced by:  seqomeq12  7055  seqeq3  12034  trpredeq1  29504  trpredeq2  29505  trpred0  29520
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