Theorem wrecseq2 8303

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

Assertion

Ref	Expression
wrecseq2	⊢ (𝐴 = 𝐵 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐵, 𝐹))

Proof of Theorem wrecseq2

Step	Hyp	Ref	Expression
1		eqid 2732	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2732	. 2 ⊢ 𝐹 = 𝐹
3		wrecseq123 8298	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐹) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐵, 𝐹))
4	1, 2, 3	mp3an13 1452	1 ⊢ (𝐴 = 𝐵 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐵, 𝐹))

Syntax hints:   → wi 4   = wceq 1541  wrecscwrecs 8295

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fv 6551  df-ov 7411  df-frecs 8265  df-wrecs 8296

This theorem is referenced by:  csbrecsg  36204