Theorem wrecseq2 7563

Description: Equality theorem for the well-founded 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 2771	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2771	. 2 ⊢ 𝐹 = 𝐹
3		wrecseq123 7560	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐹) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐵, 𝐹))
4	1, 2, 3	mp3an13 1563	1 ⊢ (𝐴 = 𝐵 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐵, 𝐹))

Syntax hints:   → wi 4   = wceq 1631  wrecscwrecs 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-iota 5994  df-fv 6039  df-wrecs 7559

This theorem is referenced by:  csbrecsg  33511