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

Theorem 2reuswap 3136
 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:   ,,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem 2reuswap
StepHypRef Expression
1 df-rmo 2713 . . 3
21ralbii 2729 . 2
3 df-ral 2710 . . . 4
4 moanimv 2339 . . . . 5
54albii 1575 . . . 4
63, 5bitr4i 244 . . 3
7 2euswap 2357 . . . 4
8 df-reu 2712 . . . . 5
9 r19.42v 2862 . . . . . . . 8
10 df-rex 2711 . . . . . . . 8
119, 10bitr3i 243 . . . . . . 7
12 an12 773 . . . . . . . 8
1312exbii 1592 . . . . . . 7
1411, 13bitri 241 . . . . . 6
1514eubii 2290 . . . . 5
168, 15bitri 241 . . . 4
17 df-reu 2712 . . . . 5
18 r19.42v 2862 . . . . . . 7
19 df-rex 2711 . . . . . . 7
2018, 19bitr3i 243 . . . . . 6
2120eubii 2290 . . . . 5
2217, 21bitri 241 . . . 4
237, 16, 223imtr4g 262 . . 3
246, 23sylbi 188 . 2
252, 24sylbi 188 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549  wex 1550   wcel 1725  weu 2281  wmo 2282  wral 2705  wrex 2706  wreu 2707  wrmo 2708 This theorem is referenced by:  reuxfr2d  4746  reuxfr3d  23976 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-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713
 Copyright terms: Public domain W3C validator