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Theorem 2eu4 2196
 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 2193 for a condition under which the naive definition holds and 2exeu 2190 for a one-way implication. See 2eu5 2197 and 2eu8 2200 for alternate definitions. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu4
Distinct variable groups:   ,,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem 2eu4
StepHypRef Expression
1 nfv 1629 . . . 4
21eu3 2139 . . 3
3 nfv 1629 . . . 4
43eu3 2139 . . 3
52, 4anbi12i 681 . 2
6 an4 800 . 2
7 excom 1765 . . . . 5
87anbi2i 678 . . . 4
9 anidm 628 . . . 4
108, 9bitri 242 . . 3
11 19.26 1592 . . . . . . . 8
12 nfa1 1719 . . . . . . . . . . 11
131219.3 1760 . . . . . . . . . 10
1413anbi2i 678 . . . . . . . . 9
15 jcab 836 . . . . . . . . . . . . 13
1615albii 1554 . . . . . . . . . . . 12
17 19.26 1592 . . . . . . . . . . . 12
1816, 17bitri 242 . . . . . . . . . . 11
1918albii 1554 . . . . . . . . . 10
20 19.26 1592 . . . . . . . . . 10
2119, 20bitri 242 . . . . . . . . 9
2214, 21bitr4i 245 . . . . . . . 8
2311, 22bitr2i 243 . . . . . . 7
24 19.26 1592 . . . . . . . . 9
25 nfa1 1719 . . . . . . . . . . 11
262519.3 1760 . . . . . . . . . 10
27 alcom 1568 . . . . . . . . . 10
2826, 27anbi12i 681 . . . . . . . . 9
2924, 28bitri 242 . . . . . . . 8
3029albii 1554 . . . . . . 7
3123, 30bitr4i 245 . . . . . 6
32 19.23v 2021 . . . . . . . 8
33 19.23v 2021 . . . . . . . 8
3432, 33anbi12i 681 . . . . . . 7
35342albii 1555 . . . . . 6
36 nfe1 1566 . . . . . . . 8
37 nfv 1629 . . . . . . . 8
3836, 37nfim 1735 . . . . . . 7
39 nfe1 1566 . . . . . . . 8
40 nfv 1629 . . . . . . . 8
4139, 40nfim 1735 . . . . . . 7
4238, 41aaan 1811 . . . . . 6
4331, 35, 423bitri 264 . . . . 5
44432exbii 1581 . . . 4
45 eeanv 2055 . . . 4
4644, 45bitr2i 243 . . 3
4710, 46anbi12i 681 . 2
485, 6, 473bitri 264 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wa 360  wal 1532  wex 1537   wceq 1619  weu 2114 This theorem is referenced by:  2eu5  2197  2eu6  2198 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118
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