Theorem otth2

Description: Ordered triple theorem, with triple expressed with ordered pairs. (Contributed by NM, 1-May-1995) (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	otth2	$$⊢⟨⟨A,B⟩,R⟩=⟨⟨C,D⟩,S⟩\leftrightarrow \left(A=C\wedge B=D\wedge R=S\right)$$

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	1 2	opth	$$⊢⟨A,B⟩=⟨C,D⟩\leftrightarrow \left(A=C\wedge B=D\right)$$
5	4	anbi1i	$$⊢\left(⟨A,B⟩=⟨C,D⟩\wedge R=S\right)\leftrightarrow \left(\left(A=C\wedge B=D\right)\wedge R=S\right)$$
6		opex	$$⊢⟨A,B⟩\in \mathrm{V}$$
7	6 3	opth	$$⊢⟨⟨A,B⟩,R⟩=⟨⟨C,D⟩,S⟩\leftrightarrow \left(⟨A,B⟩=⟨C,D⟩\wedge R=S\right)$$
8		df-3an	$$⊢\left(A=C\wedge B=D\wedge R=S\right)\leftrightarrow \left(\left(A=C\wedge B=D\right)\wedge R=S\right)$$
9	5 7 8	3bitr4i	$$⊢⟨⟨A,B⟩,R⟩=⟨⟨C,D⟩,S⟩\leftrightarrow \left(A=C\wedge B=D\wedge R=S\right)$$