Theorem 2eu5 2288

Description: An alternate definition of double existential uniqueness (see 2eu4 2287). A mistake sometimes made in the literature is to use E!xE!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.xE*yph 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!xE!yph /\ A.xE*yph) <-> (E.xE.yph /\ E.zE.wA.xA.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 2284	. . 3 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
2	1	pm5.32ri 619	. 2 |- ((E!xE!yph /\ A.xE*yph) <-> ((E!xE.yph /\ E!yE.xph) /\ A.xE*yph))
3		eumo 2244	. . . . 5 |- (E!yE.xph -> E*yE.xph)
4	3	adantl 452	. . . 4 |- ((E!xE.yph /\ E!yE.xph) -> E*yE.xph)
5		2moex 2275	. . . 4 |- (E*yE.xph -> A.xE*yph)
6	4, 5	syl 15	. . 3 |- ((E!xE.yph /\ E!yE.xph) -> A.xE*yph)
7	6	pm4.71i 613	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> ((E!xE.yph /\ E!yE.xph) /\ A.xE*yph))
8		2eu4 2287	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
9	2, 7, 8	3bitr2i 264	1 |- ((E!xE!yph /\ A.xE*yph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541  E!weu 2204  E*wmo 2205

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  2reu5lem3  3043