Theorem wrecseq123 8324

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 5835	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 ∘ 2nd ) = (𝐺 ∘ 2nd ))
2		frecseq123 8293	. . 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 8323	. 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
5		df-wrecs 8323	. 2 ⊢ wrecs(𝑆, 𝐵, 𝐺) = frecs(𝑆, 𝐵, (𝐺 ∘ 2nd ))
6	3, 4, 5	3eqtr4g 2821	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∘ ccom 5655  2^nd c2nd 7998  frecscfrecs 8291  wrecscwrecs 8322

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-iota 6493  df-fv 6545  df-ov 7421  df-frecs 8292  df-wrecs 8323

This theorem is used by:  wrecseq1  8326  wrecseq2  8327  wrecseq3  8328