Theorem wrecseq1 8294

Description: Equality theorem for the well-ordered recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)

Assertion

Ref	Expression
wrecseq1	⊢ (𝑅 = 𝑆 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐴, 𝐹))

Proof of Theorem wrecseq1

Step	Hyp	Ref	Expression
1		eqid 2729	. 2 ⊢ 𝐴 = 𝐴
2		eqid 2729	. 2 ⊢ 𝐹 = 𝐹
3		wrecseq123 8292	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐴 ∧ 𝐹 = 𝐹) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐴, 𝐹))
4	1, 2, 3	mp3an23 1455	1 ⊢ (𝑅 = 𝑆 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐴, 𝐹))

Syntax hints:   → wi 4   = wceq 1540  wrecscwrecs 8290

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-iota 6464  df-fv 6519  df-ov 7390  df-frecs 8260  df-wrecs 8291

This theorem is referenced by:  csbrecsg  37316