Theorem altopthbg

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012)

	Ref	Expression
Assertion	altopthbg	$$⊢\left(A\in V\wedge D\in W\right)\to \left(⟪A,B⟫=⟪C,D⟫\leftrightarrow \left(A=C\wedge B=D\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		altopthsn	$$⊢⟪A,B⟫=⟪C,D⟫\leftrightarrow \left(\left\{A\right\}=\left\{C\right\}\wedge \left\{B\right\}=\left\{D\right\}\right)$$
2		sneqbg	$$⊢A\in V\to \left(\left\{A\right\}=\left\{C\right\}\leftrightarrow A=C\right)$$
3		sneqbg	$$⊢D\in W\to \left(\left\{D\right\}=\left\{B\right\}\leftrightarrow D=B\right)$$
4		eqcom	$$⊢\left\{B\right\}=\left\{D\right\}\leftrightarrow \left\{D\right\}=\left\{B\right\}$$
5		eqcom	$$⊢B=D\leftrightarrow D=B$$
6	3 4 5	3bitr4g	$$⊢D\in W\to \left(\left\{B\right\}=\left\{D\right\}\leftrightarrow B=D\right)$$
7	2 6	bi2anan9	$$⊢\left(A\in V\wedge D\in W\right)\to \left(\left(\left\{A\right\}=\left\{C\right\}\wedge \left\{B\right\}=\left\{D\right\}\right)\leftrightarrow \left(A=C\wedge B=D\right)\right)$$
8	1 7	syl5bb	$$⊢\left(A\in V\wedge D\in W\right)\to \left(⟪A,B⟫=⟪C,D⟫\leftrightarrow \left(A=C\wedge B=D\right)\right)$$