Theorem wrecseq3 8260

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

Assertion

Ref	Expression
wrecseq3	⊢ (𝐹 = 𝐺 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐴, 𝐺))

Proof of Theorem wrecseq3

Step	Hyp	Ref	Expression
1		eqid 2737	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2737	. 2 ⊢ 𝐴 = 𝐴
3		wrecseq123 8256	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐴, 𝐺))
4	1, 2, 3	mp3an12 1454	1 ⊢ (𝐹 = 𝐺 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐴, 𝐺))

Syntax hints:   → wi 4   = wceq 1542  wrecscwrecs 8254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fv 6500  df-ov 7363  df-frecs 8224  df-wrecs 8255

This theorem is referenced by:  recseq  8306  bpolylem  16004