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

Theorem 2reuswap 3302
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
2reuswap
Distinct variable groups:   , ,   ,

Proof of Theorem 2reuswap
StepHypRef Expression
1 df-rmo 2815 . . 3
21ralbii 2888 . 2
3 df-ral 2812 . . . 4
4 moanimv 2352 . . . . 5
54albii 1640 . . . 4
63, 5bitr4i 252 . . 3
7 2euswap 2370 . . . 4
8 df-reu 2814 . . . . 5
9 r19.42v 3012 . . . . . . . 8
10 df-rex 2813 . . . . . . . 8
119, 10bitr3i 251 . . . . . . 7
12 an12 797 . . . . . . . 8
1312exbii 1667 . . . . . . 7
1411, 13bitri 249 . . . . . 6
1514eubii 2306 . . . . 5
168, 15bitri 249 . . . 4
17 df-reu 2814 . . . . 5
18 r19.42v 3012 . . . . . . 7
19 df-rex 2813 . . . . . . 7
2018, 19bitr3i 251 . . . . . 6
2120eubii 2306 . . . . 5
2217, 21bitri 249 . . . 4
237, 16, 223imtr4g 270 . . 3
246, 23sylbi 195 . 2
252, 24sylbi 195 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  E.wex 1612  e.wcel 1818  E!weu 2282  E*wmo 2283  A.wral 2807  E.wrex 2808  E!wreu 2809  E*wrmo 2810
This theorem is referenced by:  reuxfr2d  4675  reuxfr3d  27388
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-eu 2286  df-mo 2287  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815
  Copyright terms: Public domain W3C validator