Theorem wrecseq123 8318

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 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Proof of Theorem wrecseq123

Step	Hyp	Ref	Expression
1		coeq1 5854	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 ∘ 2nd ) = (𝐺 ∘ 2nd ))
2		frecseq123 8286	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ (𝐹 ∘ 2nd ) = (𝐺 ∘ 2nd )) → frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd )))
3	1, 2	syl3an3 1162	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd )))
4		df-wrecs 8316	. 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
5		df-wrecs 8316	. 2 ⊢ wrecs(𝑆, 𝐵, 𝐺) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd ))
6	3, 4, 5	3eqtr4g 2790	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∘ ccom 5676  2^nd c2nd 7990  frecscfrecs 8284  wrecscwrecs 8315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-iota 6495  df-fv 6551  df-ov 7419  df-frecs 8285  df-wrecs 8316

This theorem is referenced by:  wrecseq1  8322  wrecseq2  8323  wrecseq3  8324