Metamath Proof Explorer


Theorem 2eu5

Description: An alternate definition of double existential uniqueness (see 2eu4 ). A mistake sometimes made in the literature is to use E! x E! y to mean "exactly one x and exactly one y ". (For example, see Proposition 7.53 of TakeutiZaring p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining A. x E* y ph as an additional condition. The correct definition apparently has never been published. ( E* means "there exists at most one".) (Contributed by NM, 26-Oct-2003) Avoid ax-13 . (Revised by Wolf Lammen, 2-Oct-2023)

Ref Expression
Assertion 2eu5 ∃!x∃!yφx*yφxyφzwxyφx=zy=w

Proof

Step Hyp Ref Expression
1 2eu1v x*yφ∃!x∃!yφ∃!xyφ∃!yxφ
2 1 pm5.32ri ∃!x∃!yφx*yφ∃!xyφ∃!yxφx*yφ
3 eumo ∃!yxφ*yxφ
4 2moexv *yxφx*yφ
5 3 4 syl ∃!yxφx*yφ
6 5 adantl ∃!xyφ∃!yxφx*yφ
7 6 pm4.71i ∃!xyφ∃!yxφ∃!xyφ∃!yxφx*yφ
8 2eu4 ∃!xyφ∃!yxφxyφzwxyφx=zy=w
9 2 7 8 3bitr2i ∃!x∃!yφx*yφxyφzwxyφx=zy=w