Theorem wrecseq123

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)

	Ref	Expression
Assertion	wrecseq123	|- ( ( R = S /\ A = B /\ F = G ) -> wrecs ( R , A , F ) = wrecs ( S , B , G ) )

Proof

Step	Hyp	Ref	Expression
1		coeq1	|- ( F = G -> ( F o. 2nd ) = ( G o. 2nd ) )
2		frecseq123	|- ( ( R = S /\ A = B /\ ( F o. 2nd ) = ( G o. 2nd ) ) -> frecs ( R , A , ( F o. 2nd ) ) = frecs ( S , B , ( G o. 2nd ) ) )
3	1 2	syl3an3	|- ( ( R = S /\ A = B /\ F = G ) -> frecs ( R , A , ( F o. 2nd ) ) = frecs ( S , B , ( G o. 2nd ) ) )
4		df-wrecs	|- wrecs ( R , A , F ) = frecs ( R , A , ( F o. 2nd ) )
5		df-wrecs	|- wrecs ( S , B , G ) = frecs ( S , B , ( G o. 2nd ) )
6	3 4 5	3eqtr4g	|- ( ( R = S /\ A = B /\ F = G ) -> wrecs ( R , A , F ) = wrecs ( S , B , G ) )