Theorem rdgeq12 7509

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

Step	Hyp	Ref	Expression
1		rdgeq2 7508	. 2 |- ( A = B -> rec ( F , A ) = rec ( F , B ) )
2		rdgeq1 7507	. 2 |- ( F = G -> rec ( F , B ) = rec ( G , B ) )
3	1, 2	sylan9eqr 2678	1 |- ( ( F = G /\ A = B ) -> rec ( F , A ) = rec ( G , B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   reccrdg 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  seqomeq12  7549  seqeq3  12806  trpredeq1  31720  trpredeq2  31721  trpred0  31736  csbfinxpg  33225