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Theorem nfwrecs 8303
Description: Bound-variable hypothesis builder for the well-ordered recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
Hypotheses
Ref Expression
nfwrecs.1 𝑥𝑅
nfwrecs.2 𝑥𝐴
nfwrecs.3 𝑥𝐹
Assertion
Ref Expression
nfwrecs 𝑥wrecs(𝑅, 𝐴, 𝐹)

Proof of Theorem nfwrecs
StepHypRef Expression
1 df-wrecs 8299 . 2 wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
2 nfwrecs.1 . . 3 𝑥𝑅
3 nfwrecs.2 . . 3 𝑥𝐴
4 nfwrecs.3 . . . 4 𝑥𝐹
5 nfcv 2901 . . . 4 𝑥2nd
64, 5nfco 5864 . . 3 𝑥(𝐹 ∘ 2nd )
72, 3, 6nffrecs 8270 . 2 𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
81, 7nfcxfr 2899 1 𝑥wrecs(𝑅, 𝐴, 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2881  ccom 5679  2nd c2nd 7976  frecscfrecs 8267  wrecscwrecs 8298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-iota 6494  df-fv 6550  df-ov 7414  df-frecs 8268  df-wrecs 8299
This theorem is referenced by:  nfrecs  8377
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