Theorem mpoeq123i

Description: An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013)

	Ref	Expression
Hypotheses	mpoeq123i.1	$$⊢A=D$$
	mpoeq123i.2	$$⊢B=E$$
	mpoeq123i.3	$$⊢C=F$$
Assertion	mpoeq123i	$$⊢\left(x\in A,y\in B⟼C\right)=\left(x\in D,y\in E⟼F\right)$$

Proof

Step	Hyp	Ref	Expression
1		mpoeq123i.1	$$⊢A=D$$
2		mpoeq123i.2	$$⊢B=E$$
3		mpoeq123i.3	$$⊢C=F$$
4	1	a1i	$$⊢\top \to A=D$$
5	2	a1i	$$⊢\top \to B=E$$
6	3	a1i	$$⊢\top \to C=F$$
7	4 5 6	mpoeq123dv	$$⊢\top \to \left(x\in A,y\in B⟼C\right)=\left(x\in D,y\in E⟼F\right)$$
8	7	mptru	$$⊢\left(x\in A,y\in B⟼C\right)=\left(x\in D,y\in E⟼F\right)$$