Theorem unceq

Description: Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021)

	Ref	Expression
Assertion	unceq	$$⊢A=B\to uncurryA=uncurryB$$

Proof

Step	Hyp	Ref	Expression
1		fveq1	$$⊢A=B\to A\left(x\right)=B\left(x\right)$$
2	1	breqd	$$⊢A=B\to \left(yA\left(x\right)z\leftrightarrow yB\left(x\right)z\right)$$
3	2	oprabbidv	$$⊢A=B\to \left\{⟨⟨x,y⟩,z⟩|yA\left(x\right)z\right\}=\left\{⟨⟨x,y⟩,z⟩|yB\left(x\right)z\right\}$$
4		df-unc	$$⊢uncurryA=\left\{⟨⟨x,y⟩,z⟩|yA\left(x\right)z\right\}$$
5		df-unc	$$⊢uncurryB=\left\{⟨⟨x,y⟩,z⟩|yB\left(x\right)z\right\}$$
6	3 4 5	3eqtr4g	$$⊢A=B\to uncurryA=uncurryB$$