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Theorem 2reu5lem3 3043
Description: Lemma for 2reu5 3044. 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 3131. (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 3041 . . 3
2 2reu5lem2 3042 . . 3
31, 2anbi12i 678 . 2
4 2eu5 2288 . 2
5 3anass 938 . . . . . . 7
65exbii 1582 . . . . . 6
7 19.42v 1905 . . . . . 6
8 df-rex 2620 . . . . . . . 8
98bicomi 193 . . . . . . 7
109anbi2i 675 . . . . . 6
116, 7, 103bitri 262 . . . . 5
1211exbii 1582 . . . 4
13 df-rex 2620 . . . 4
1412, 13bitr4i 243 . . 3
15 3anan12 947 . . . . . . . . . . 11
1615imbi1i 315 . . . . . . . . . 10
17 impexp 433 . . . . . . . . . 10
18 impexp 433 . . . . . . . . . . 11
1918imbi2i 303 . . . . . . . . . 10
2016, 17, 193bitri 262 . . . . . . . . 9
2120albii 1566 . . . . . . . 8
22 df-ral 2619 . . . . . . . 8
23 r19.21v 2701 . . . . . . . 8
2421, 22, 233bitr2i 264 . . . . . . 7
2524albii 1566 . . . . . 6
26 df-ral 2619 . . . . . 6
2725, 26bitr4i 243 . . . . 5
2827exbii 1582 . . . 4
2928exbii 1582 . . 3
3014, 29anbi12i 678 . 2
313, 4, 303bitri 262 1
Colors of variables: wff setvar class
Syntax hints:   wi 4   wb 176   wa 358   w3a 934  wal 1540  wex 1541   wcel 1710  weu 2204  wmo 2205  wral 2614  wrex 2615  wreu 2616  wrmo 2617
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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622
This theorem is referenced by:  2reu5  3044
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