Theorem wrecseq123 8306

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 5843	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 ∘ 2nd ) = (𝐺 ∘ 2nd ))
2		frecseq123 8275	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ (𝐹 ∘ 2nd ) = (𝐺 ∘ 2nd )) → frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd )))
3	1, 2	syl3an3 1183	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd )))
4		df-wrecs 8305	. 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
5		df-wrecs 8305	. 2 ⊢ wrecs(𝑆, 𝐵, 𝐺) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd ))
6	3, 4, 5	3eqtr4g 2823	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∘ ccom 5665  2^nd c2nd 7981  frecscfrecs 8273  wrecscwrecs 8304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-iota 6492  df-fv 6544  df-ov 7413  df-frecs 8274  df-wrecs 8305

This theorem is referenced by:  wrecseq1  8308  wrecseq2  8309  wrecseq3  8310