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Theorem 2eu1 2284
 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 2242 . . . . . . . 8
2 eu5 2242 . . . . . . . . . 10
32exbii 1582 . . . . . . . . 9
42mobii 2240 . . . . . . . . 9
53, 4anbi12i 678 . . . . . . . 8
61, 5bitri 240 . . . . . . 7
76simprbi 450 . . . . . 6
8 sp 1747 . . . . . . . . . . . 12
98anim2i 552 . . . . . . . . . . 11
109ancoms 439 . . . . . . . . . 10
1110moimi 2251 . . . . . . . . 9
12 nfa1 1788 . . . . . . . . . 10
1312moanim 2260 . . . . . . . . 9
1411, 13sylib 188 . . . . . . . 8
1514ancrd 537 . . . . . . 7
16 2moswap 2279 . . . . . . . . 9
1716com12 27 . . . . . . . 8
1817imdistani 671 . . . . . . 7
1915, 18syl6 29 . . . . . 6
207, 19syl 15 . . . . 5
21 2eu2ex 2278 . . . . . 6
22 excom 1741 . . . . . . 7
2321, 22sylib 188 . . . . . 6
2421, 23jca 518 . . . . 5
2520, 24jctild 527 . . . 4
26 eu5 2242 . . . . . 6
27 eu5 2242 . . . . . 6
2826, 27anbi12i 678 . . . . 5
29 an4 797 . . . . 5
3028, 29bitri 240 . . . 4
3125, 30syl6ibr 218 . . 3
3231com12 27 . 2
33 2exeu 2281 . 2
3432, 33impbid1 194 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wex 1541  weu 2204  wmo 2205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  2eu2  2285  2eu3  2286  2eu5  2288
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