Theorem opabidw

Description: The law of concretion. Special case of Theorem 9.5 of Quine p. 61. Version of opabid with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 14-Apr-1995) (Revised by Gino Giotto, 26-Jan-2024)

	Ref	Expression
Assertion	opabidw	|- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )

Proof

Step	Hyp	Ref	Expression
1		opex	|- <. x , y >. e. _V
2		copsexgw	|- ( z = <. x , y >. -> ( ph <-> E. x E. y ( z = <. x , y >. /\ ph ) ) )
3	2	bicomd	|- ( z = <. x , y >. -> ( E. x E. y ( z = <. x , y >. /\ ph ) <-> ph ) )
4		df-opab	|- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
5	1 3 4	elab2	|- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )