Theorem 2sb8e

Description: An equivalent expression for double existence. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring more disjoint variables, but fewer axioms, see 2sb8ev . (Contributed by Wolf Lammen, 2-Nov-2019) (New usage is discouraged.)

	Ref	Expression
Assertion	2sb8e	|- ( E. x E. y ph <-> E. z E. w [ z / x ] [ w / y ] ph )

Proof

Step	Hyp	Ref	Expression
1		nfv	|- F/ w ph
2	1	sb8e	|- ( E. y ph <-> E. w [ w / y ] ph )
3	2	exbii	|- ( E. x E. y ph <-> E. x E. w [ w / y ] ph )
4		excom	|- ( E. x E. w [ w / y ] ph <-> E. w E. x [ w / y ] ph )
5	3 4	bitri	|- ( E. x E. y ph <-> E. w E. x [ w / y ] ph )
6		nfv	|- F/ z ph
7	6	nfsb	|- F/ z [ w / y ] ph
8	7	sb8e	|- ( E. x [ w / y ] ph <-> E. z [ z / x ] [ w / y ] ph )
9	8	exbii	|- ( E. w E. x [ w / y ] ph <-> E. w E. z [ z / x ] [ w / y ] ph )
10		excom	|- ( E. w E. z [ z / x ] [ w / y ] ph <-> E. z E. w [ z / x ] [ w / y ] ph )
11	5 9 10	3bitri	|- ( E. x E. y ph <-> E. z E. w [ z / x ] [ w / y ] ph )