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Theorem wfr2 25547
 Description: The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of at any is recursively applied to all "previous" values of . (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfr2.1
wfr2.2 Se
wfr2.3 wrecs
Assertion
Ref Expression
wfr2

Proof of Theorem wfr2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . 3
2 predeq3 25435 . . . . 5
32reseq2d 5138 . . . 4
43fveq2d 5724 . . 3
51, 4eqeq12d 2449 . 2
6 wfr2.1 . . . . 5
7 wfr2.2 . . . . 5 Se
8 wfr2.3 . . . . 5 wrecs
9 eqid 2435 . . . . 5
106, 7, 8, 9wfrlem16 25545 . . . 4
1110eleq2i 2499 . . 3
126, 7, 8wfrlem12 25541 . . 3
1311, 12sylbir 205 . 2
145, 13vtoclga 3009 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725   cun 3310  csn 3806  cop 3809   Se wse 4531   wwe 4532   cdm 4870   cres 4872  cfv 5446  cpred 25430  wrecscwrecs 25522 This theorem is referenced by:  wfr3  25548  tfrALTlem  25549  tfr2ALT  25551  bpolylem  26086 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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-pred 25431  df-wrecs 25523
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