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Theorem 2reu5 3144
 Description: Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2367 and reu3 3126. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5
Distinct variable groups:   ,,,,   ,   ,,,   ,,   ,   ,
Allowed substitution hints:   (,)

Proof of Theorem 2reu5
StepHypRef Expression
1 r19.29r 2849 . . . . . . . 8
2 r19.29r 2849 . . . . . . . . 9
32reximi 2815 . . . . . . . 8
4 pm3.35 572 . . . . . . . . . 10
54reximi 2815 . . . . . . . . 9
65reximi 2815 . . . . . . . 8
7 eleq1 2498 . . . . . . . . . . . . 13
8 eleq1 2498 . . . . . . . . . . . . 13
97, 8bi2anan9 845 . . . . . . . . . . . 12
109biimpac 474 . . . . . . . . . . 11
1110ancomd 440 . . . . . . . . . 10
1211ex 425 . . . . . . . . 9
1312rexlimivv 2837 . . . . . . . 8
141, 3, 6, 134syl 20 . . . . . . 7
1514ex 425 . . . . . 6
1615pm4.71rd 618 . . . . 5
17 anass 632 . . . . 5
1816, 17syl6bb 254 . . . 4
19182exbidv 1639 . . 3
2019pm5.32i 620 . 2
21 2reu5lem3 3143 . 2
22 df-rex 2713 . . . 4
23 r19.42v 2864 . . . . . 6
24 df-rex 2713 . . . . . 6
2523, 24bitr3i 244 . . . . 5
2625exbii 1593 . . . 4
2722, 26bitri 242 . . 3
2827anbi2i 677 . 2
2920, 21, 283bitr4i 270 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551   wcel 1726  wral 2707  wrex 2708  wreu 2709  wrmo 2710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-cleq 2431  df-clel 2434  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715
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