Metamath Proof Explorer


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 2sb8ef . (Contributed by Wolf Lammen, 2-Nov-2019) (New usage is discouraged.)

Ref Expression
Assertion 2sb8e xyφzwzxwyφ

Proof

Step Hyp Ref Expression
1 nfv wφ
2 1 sb8e yφwwyφ
3 2 exbii xyφxwwyφ
4 excom xwwyφwxwyφ
5 3 4 bitri xyφwxwyφ
6 nfv zφ
7 6 nfsb zwyφ
8 7 sb8e xwyφzzxwyφ
9 8 exbii wxwyφwzzxwyφ
10 excom wzzxwyφzwzxwyφ
11 5 9 10 3bitri xyφzwzxwyφ