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Theorem 2eu1 2193
 Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu1

Proof of Theorem 2eu1
StepHypRef Expression
1 eu5 2151 . . . . . . . 8
2 eu5 2151 . . . . . . . . . 10
32exbii 1580 . . . . . . . . 9
42mobii 2149 . . . . . . . . 9
53, 4anbi12i 681 . . . . . . . 8
61, 5bitri 242 . . . . . . 7
76simprbi 452 . . . . . 6
8 ax-4 1692 . . . . . . . . . . . 12
98anim2i 555 . . . . . . . . . . 11
109ancoms 441 . . . . . . . . . 10
1110immoi 2160 . . . . . . . . 9
12 nfa1 1719 . . . . . . . . . 10
1312moanim 2169 . . . . . . . . 9
1411, 13sylib 190 . . . . . . . 8
1514ancrd 539 . . . . . . 7
16 2moswap 2188 . . . . . . . . 9
1716com12 29 . . . . . . . 8
1817imdistani 674 . . . . . . 7
1915, 18syl6 31 . . . . . 6
207, 19syl 17 . . . . 5
21 2eu2ex 2187 . . . . . 6
22 excom 1765 . . . . . . 7
2321, 22sylib 190 . . . . . 6
2421, 23jca 520 . . . . 5
2520, 24jctild 529 . . . 4
26 eu5 2151 . . . . . 6
27 eu5 2151 . . . . . 6
2826, 27anbi12i 681 . . . . 5
29 an4 800 . . . . 5
3028, 29bitri 242 . . . 4
3125, 30syl6ibr 220 . . 3
3231com12 29 . 2
33 2exeu 2190 . 2
3432, 33impbid1 196 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wa 360  wal 1532  wex 1537  weu 2114  wmo 2115 This theorem is referenced by:  2eu2  2194  2eu3  2195  2eu5  2197 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  df-mo 2119
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