Theorem 2eu4 2380

Description: This theorem provides us with a definition of double existential uniqueness ("exactly one and exactly one "). Naively one might think (incorrectly) that it could be defined by . See 2eu1 2376 for a condition under which the naive definition holds and 2exeu 2371 for a one-way implication. See 2eu5 2382 and 2eu8 2386 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)

Assertion

Ref	Expression
2eu4

Distinct variable groups:   ,,,   ,,

Proof of Theorem 2eu4

Step	Hyp	Ref	Expression
1		eu5 2310	. . 3
2		eu5 2310	. . . 4
3		excom 1849	. . . . 5
4	3	anbi1i 695	. . . 4
5	2, 4	bitri 249	. . 3
6	1, 5	anbi12i 697	. 2
7		anandi 828	. 2
8		2mo2 2372	. . 3
9	8	anbi2i 694	. 2
10	6, 7, 9	3bitr2i 273	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  E!weu 2282  E*wmo 2283

This theorem is referenced by:  2eu5  2382  2eu6  2383  2eu6OLD  2384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617  df-eu 2286  df-mo 2287