Theorem ssopab2b

Description: Equivalence of ordered pair abstraction subclass and implication. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker ssopab2bw when possible. (Contributed by NM, 27-Dec-1996) (Proof shortened by Mario Carneiro, 18-Nov-2016) (New usage is discouraged.)

	Ref	Expression
Assertion	ssopab2b	|- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } <-> A. x A. y ( ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		nfopab1	|- F/_ x { <. x , y >. | ph }
2		nfopab1	|- F/_ x { <. x , y >. | ps }
3	1 2	nfss	|- F/ x { <. x , y >. | ph } C_ { <. x , y >. | ps }
4		nfopab2	|- F/_ y { <. x , y >. | ph }
5		nfopab2	|- F/_ y { <. x , y >. | ps }
6	4 5	nfss	|- F/ y { <. x , y >. | ph } C_ { <. x , y >. | ps }
7		ssel	|- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } -> ( <. x , y >. e. { <. x , y >. | ph } -> <. x , y >. e. { <. x , y >. | ps } ) )
8		opabid	|- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
9		opabid	|- ( <. x , y >. e. { <. x , y >. | ps } <-> ps )
10	7 8 9	3imtr3g	|- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } -> ( ph -> ps ) )
11	6 10	alrimi	|- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } -> A. y ( ph -> ps ) )
12	3 11	alrimi	|- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } -> A. x A. y ( ph -> ps ) )
13		ssopab2	|- ( A. x A. y ( ph -> ps ) -> { <. x , y >. | ph } C_ { <. x , y >. | ps } )
14	12 13	impbii	|- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } <-> A. x A. y ( ph -> ps ) )