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Metamath Proof Explorer

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	$⊢ (R = S \land A = B \land F = G) \to wrecs (R, A, F) = wrecs (S, B, G)$

Proof

Step	Hyp	Ref	Expression
1		coeq1	$⊢ F = G \to F \circ 2^{nd} = G \circ 2^{nd}$
2		frecseq123	$⊢ (R = S \land A = B \land F \circ 2^{nd} = G \circ 2^{nd}) \to frecs (R, A, F \circ 2^{nd}) = frecs (S, B, G \circ 2^{nd})$
3	1 2	syl3an3	$⊢ (R = S \land A = B \land F = G) \to frecs (R, A, F \circ 2^{nd}) = frecs (S, B, G \circ 2^{nd})$
4		df-wrecs	$⊢ wrecs (R, A, F) = frecs (R, A, F \circ 2^{nd})$
5		df-wrecs	$⊢ wrecs (S, B, G) = frecs (S, B, G \circ 2^{nd})$
6	3 4 5	3eqtr4g	$⊢ (R = S \land A = B \land F = G) \to wrecs (R, A, F) = wrecs (S, B, G)$