Theorem wrecseq123 8294

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 5829	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 ∘ 2nd ) = (𝐺 ∘ 2nd ))
2		frecseq123 8263	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ (𝐹 ∘ 2nd ) = (𝐺 ∘ 2nd )) → frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd )))
3	1, 2	syl3an3 1178	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd )))
4		df-wrecs 8293	. 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
5		df-wrecs 8293	. 2 ⊢ wrecs(𝑆, 𝐵, 𝐺) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd ))
6	3, 4, 5	3eqtr4g 2822	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Syntax hints:   → wi 4   ∧ w3a 1098   = wceq 1560   ∘ ccom 5651  2^nd c2nd 7969  frecscfrecs 8261  wrecscwrecs 8292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-iota 6477  df-fv 6529  df-ov 7399  df-frecs 8262  df-wrecs 8293

This theorem is referenced by:  wrecseq1  8296  wrecseq2  8297  wrecseq3  8298