Theorem wrecseq1 7410

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

Assertion

Ref	Expression
wrecseq1	|- ( R = S -> wrecs ( R , A , F ) = wrecs ( S , A , F ) )

Proof of Theorem wrecseq1

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- A = A
2		eqid 2622	. 2 |- F = F
3		wrecseq123 7408	. 2 |- ( ( R = S /\ A = A /\ F = F ) -> wrecs ( R , A , F ) = wrecs ( S , A , F ) )
4	1, 2, 3	mp3an23 1416	1 |- ( R = S -> wrecs ( R , A , F ) = wrecs ( S , A , F ) )

Syntax hints:   -> wi 4   = wceq 1483  wrecscwrecs 7406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  csbrecsg  33174