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Theorem wrecseq123 8339
Description: General equality theorem for the well-ordered recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
Assertion
Ref Expression
wrecseq123 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Proof of Theorem wrecseq123
StepHypRef Expression
1 coeq1 5868 . . 3 (𝐹 = 𝐺 → (𝐹 ∘ 2nd ) = (𝐺 ∘ 2nd ))
2 frecseq123 8307 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵 ∧ (𝐹 ∘ 2nd ) = (𝐺 ∘ 2nd )) → frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd )))
31, 2syl3an3 1166 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd )))
4 df-wrecs 8337 . 2 wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
5 df-wrecs 8337 . 2 wrecs(𝑆, 𝐵, 𝐺) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd ))
63, 4, 53eqtr4g 2802 1 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  ccom 5689  2nd c2nd 8013  frecscfrecs 8305  wrecscwrecs 8336
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fv 6569  df-ov 7434  df-frecs 8306  df-wrecs 8337
This theorem is referenced by:  wrecseq1  8343  wrecseq2  8344  wrecseq3  8345
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