Theorem 2eu5 2557

Description: An alternate definition of double existential uniqueness (see 2eu4 2556). 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 "exists at most one."). (Contributed by NM, 26-Oct-2003.)

Assertion

Ref	Expression
2eu5	|- ( ( E! x E! y ph /\ A. x E* y ph ) <-> ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) )

Distinct variable groups:   x, y, z, w   ph, z, w

Allowed substitution hints:   ph( x, y)

Proof of Theorem 2eu5

Step	Hyp	Ref	Expression
1		2eu1 2553	. . 3 |- ( A. x E* y ph -> ( E! x E! y ph <-> ( E! x E. y ph /\ E! y E. x ph ) ) )
2	1	pm5.32ri 670	. 2 |- ( ( E! x E! y ph /\ A. x E* y ph ) <-> ( ( E! x E. y ph /\ E! y E. x ph ) /\ A. x E* y ph ) )
3		eumo 2499	. . . . 5 |- ( E! y E. x ph -> E* y E. x ph )
4	3	adantl 482	. . . 4 |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E* y E. x ph )
5		2moex 2543	. . . 4 |- ( E* y E. x ph -> A. x E* y ph )
6	4, 5	syl 17	. . 3 |- ( ( E! x E. y ph /\ E! y E. x ph ) -> A. x E* y ph )
7	6	pm4.71i 664	. 2 |- ( ( E! x E. y ph /\ E! y E. x ph ) <-> ( ( E! x E. y ph /\ E! y E. x ph ) /\ A. x E* y ph ) )
8		2eu4 2556	. 2 |- ( ( E! x E. y ph /\ E! y E. x ph ) <-> ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) )
9	2, 7, 8	3bitr2i 288	1 |- ( ( E! x E! y ph /\ A. x E* y ph ) <-> ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   E!weu 2470   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  2reu5lem3  3415