Theorem opelopaba

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	opelopaba.1	|- A e. _V
	opelopaba.2	|- B e. _V
	opelopaba.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
Assertion	opelopaba	|- ( <. A , B >. e. { <. x , y >. | ph } <-> ps )

Proof

Step	Hyp	Ref	Expression
1		opelopaba.1	|- A e. _V
2		opelopaba.2	|- B e. _V
3		opelopaba.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
4	3	opelopabga	|- ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) )
5	1 2 4	mp2an	|- ( <. A , B >. e. { <. x , y >. | ph } <-> ps )