Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  2reu5 Unicode version

Theorem 2reu5 3308
 Description: Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2382 and reu3 3289. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5
Distinct variable groups:   ,,,,   ,   ,,,   ,,   ,   ,

Proof of Theorem 2reu5
StepHypRef Expression
1 r19.29r 2993 . . . . . . . 8
2 r19.29r 2993 . . . . . . . . 9
32reximi 2925 . . . . . . . 8
4 pm3.35 587 . . . . . . . . . 10
54reximi 2925 . . . . . . . . 9
65reximi 2925 . . . . . . . 8
7 eleq1 2529 . . . . . . . . . . . . 13
8 eleq1 2529 . . . . . . . . . . . . 13
97, 8bi2anan9 873 . . . . . . . . . . . 12
109biimpac 486 . . . . . . . . . . 11
1110ancomd 451 . . . . . . . . . 10
1211ex 434 . . . . . . . . 9
1312rexlimivv 2954 . . . . . . . 8
141, 3, 6, 134syl 21 . . . . . . 7
1514ex 434 . . . . . 6
1615pm4.71rd 635 . . . . 5
17 anass 649 . . . . 5
1816, 17syl6bb 261 . . . 4
19182exbidv 1716 . . 3
2019pm5.32i 637 . 2
21 2reu5lem3 3307 . 2
22 df-rex 2813 . . . 4
23 r19.42v 3012 . . . . . 6
24 df-rex 2813 . . . . . 6
2523, 24bitr3i 251 . . . . 5
2625exbii 1667 . . . 4
2722, 26bitri 249 . . 3
2827anbi2i 694 . 2
2920, 21, 283bitr4i 277 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808  E!wreu 2809  E*wrmo 2810 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-eu 2286  df-mo 2287  df-cleq 2449  df-clel 2452  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815
 Copyright terms: Public domain W3C validator