Theorem otth

Description: Ordered triple theorem. (Contributed by NM, 25-Sep-2014) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Hypotheses	otth.1	$$⊢A\in \mathrm{V}$$
	otth.2	$$⊢B\in \mathrm{V}$$
	otth.3	$$⊢R\in \mathrm{V}$$
Assertion	otth	$$⊢⟨A,B,R⟩=⟨C,D,S⟩\leftrightarrow \left(A=C\wedge B=D\wedge R=S\right)$$

Proof

Step	Hyp	Ref	Expression
1		otth.1	$$⊢A\in \mathrm{V}$$
2		otth.2	$$⊢B\in \mathrm{V}$$
3		otth.3	$$⊢R\in \mathrm{V}$$
4		df-ot	$$⊢⟨A,B,R⟩=⟨⟨A,B⟩,R⟩$$
5		df-ot	$$⊢⟨C,D,S⟩=⟨⟨C,D⟩,S⟩$$
6	4 5	eqeq12i	$$⊢⟨A,B,R⟩=⟨C,D,S⟩\leftrightarrow ⟨⟨A,B⟩,R⟩=⟨⟨C,D⟩,S⟩$$
7	1 2 3	otth2	$$⊢⟨⟨A,B⟩,R⟩=⟨⟨C,D⟩,S⟩\leftrightarrow \left(A=C\wedge B=D\wedge R=S\right)$$
8	6 7	bitri	$$⊢⟨A,B,R⟩=⟨C,D,S⟩\leftrightarrow \left(A=C\wedge B=D\wedge R=S\right)$$