Theorem eqopab2b

Description: Equivalence of ordered pair abstraction equality and biconditional. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker eqopab2bw when possible. (Contributed by Mario Carneiro, 4-Jan-2017) (New usage is discouraged.)

	Ref	Expression
Assertion	eqopab2b	|- ( { <. x , y >. | ph } = { <. x , y >. | ps } <-> A. x A. y ( ph <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		ssopab2b	|- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } <-> A. x A. y ( ph -> ps ) )
2		ssopab2b	|- ( { <. x , y >. | ps } C_ { <. x , y >. | ph } <-> A. x A. y ( ps -> ph ) )
3	1 2	anbi12i	|- ( ( { <. x , y >. | ph } C_ { <. x , y >. | ps } /\ { <. x , y >. | ps } C_ { <. x , y >. | ph } ) <-> ( A. x A. y ( ph -> ps ) /\ A. x A. y ( ps -> ph ) ) )
4		eqss	|- ( { <. x , y >. | ph } = { <. x , y >. | ps } <-> ( { <. x , y >. | ph } C_ { <. x , y >. | ps } /\ { <. x , y >. | ps } C_ { <. x , y >. | ph } ) )
5		2albiim	|- ( A. x A. y ( ph <-> ps ) <-> ( A. x A. y ( ph -> ps ) /\ A. x A. y ( ps -> ph ) ) )
6	3 4 5	3bitr4i	|- ( { <. x , y >. | ph } = { <. x , y >. | ps } <-> A. x A. y ( ph <-> ps ) )