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Theorem 2reu5lem3 3133
 Description: Lemma for 2reu5 3134. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3238. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5lem3
Distinct variable groups:   ,,,   ,,,   ,   ,,
Allowed substitution hints:   (,)   ()   ()

Proof of Theorem 2reu5lem3
StepHypRef Expression
1 2reu5lem1 3131 . . 3
2 2reu5lem2 3132 . . 3
31, 2anbi12i 679 . 2
4 2eu5 2364 . 2
5 3anass 940 . . . . . . 7
65exbii 1592 . . . . . 6
7 19.42v 1928 . . . . . 6
8 df-rex 2703 . . . . . . . 8
98bicomi 194 . . . . . . 7
109anbi2i 676 . . . . . 6
116, 7, 103bitri 263 . . . . 5
1211exbii 1592 . . . 4
13 df-rex 2703 . . . 4
1412, 13bitr4i 244 . . 3
15 3anan12 949 . . . . . . . . . . 11
1615imbi1i 316 . . . . . . . . . 10
17 impexp 434 . . . . . . . . . 10
18 impexp 434 . . . . . . . . . . 11
1918imbi2i 304 . . . . . . . . . 10
2016, 17, 193bitri 263 . . . . . . . . 9
2120albii 1575 . . . . . . . 8
22 df-ral 2702 . . . . . . . 8
23 r19.21v 2785 . . . . . . . 8
2421, 22, 233bitr2i 265 . . . . . . 7
2524albii 1575 . . . . . 6
26 df-ral 2702 . . . . . 6
2725, 26bitr4i 244 . . . . 5
2827exbii 1592 . . . 4
2928exbii 1592 . . 3
3014, 29anbi12i 679 . 2
313, 4, 303bitri 263 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936  wal 1549  wex 1550   wcel 1725  weu 2280  wmo 2281  wral 2697  wrex 2698  wreu 2699  wrmo 2700 This theorem is referenced by:  2reu5  3134 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705
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