Theorem opelopab

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by NM, 16-May-1995)

	Ref	Expression
Hypotheses	opelopab.1	|- A e. _V
	opelopab.2	|- B e. _V
	opelopab.3	|- ( x = A -> ( ph <-> ps ) )
	opelopab.4	|- ( y = B -> ( ps <-> ch ) )
Assertion	opelopab	|- ( <. A , B >. e. { <. x , y >. | ph } <-> ch )

Proof

Step	Hyp	Ref	Expression
1		opelopab.1	|- A e. _V
2		opelopab.2	|- B e. _V
3		opelopab.3	|- ( x = A -> ( ph <-> ps ) )
4		opelopab.4	|- ( y = B -> ( ps <-> ch ) )
5	3 4	opelopabg	|- ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )
6	1 2 5	mp2an	|- ( <. A , B >. e. { <. x , y >. | ph } <-> ch )