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Theorem freceq12 6311
 Description: Equality theorem for finite recursive function generator. (Contributed by Scott Fenton, 31-Jul-2019.)
Assertion
Ref Expression
freceq12 FRec FRec

Proof of Theorem freceq12
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 opeq2 4579 . . . . 5 0c 0c
21sneqd 3746 . . . 4 0c 0c
3 clos1eq1 5874 . . . 4 0c 0c Clos1 0c PProd 1c Clos1 0c PProd 1c
42, 3syl 15 . . 3 Clos1 0c PProd 1c Clos1 0c PProd 1c
5 pprodeq2 5835 . . . 4 PProd 1c PProd 1c
6 clos1eq2 5875 . . . 4 PProd 1c PProd 1c Clos1 0c PProd 1c Clos1 0c PProd 1c
75, 6syl 15 . . 3 Clos1 0c PProd 1c Clos1 0c PProd 1c
84, 7sylan9eqr 2407 . 2 Clos1 0c PProd 1c Clos1 0c PProd 1c
9 df-frec 6310 . 2 FRec Clos1 0c PProd 1c
10 df-frec 6310 . 2 FRec Clos1 0c PProd 1c
118, 9, 103eqtr4g 2410 1 FRec FRec
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 358   wceq 1642  cvv 2859  csn 3737  1cc1c 4134  0cc0c 4374   cplc 4375  cop 4561   cmpt 5651   PProd cpprod 5737   Clos1 cclos1 5872   FRec cfrec 6309 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-txp 5736  df-pprod 5738  df-clos1 5873  df-frec 6310 This theorem is referenced by:  frecxp  6314  frecxpg  6315
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