Theorem opelopabf

Description: The law of concretion. Theorem 9.5 of Quine p. 61. This version of opelopab uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008)

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

Proof

Step	Hyp	Ref	Expression
1		opelopabf.x	|- F/ x ps
2		opelopabf.y	|- F/ y ch
3		opelopabf.1	|- A e. _V
4		opelopabf.2	|- B e. _V
5		opelopabf.3	|- ( x = A -> ( ph <-> ps ) )
6		opelopabf.4	|- ( y = B -> ( ps <-> ch ) )
7		opelopabsb	|- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
8		nfcv	|- F/_ x B
9	8 1	nfsbcw	|- F/ x [. B / y ]. ps
10	5	sbcbidv	|- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
11	9 10	sbciegf	|- ( A e. _V -> ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. ps ) )
12	3 11	ax-mp	|- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. ps )
13	2 6	sbciegf	|- ( B e. _V -> ( [. B / y ]. ps <-> ch ) )
14	4 13	ax-mp	|- ( [. B / y ]. ps <-> ch )
15	7 12 14	3bitri	|- ( <. A , B >. e. { <. x , y >. | ph } <-> ch )