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Theorem 2reuswap 2980
 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 2564 . . 3
21ralbii 2580 . 2
3 df-ral 2561 . . . 4
4 moanimv 2214 . . . . 5
54albii 1556 . . . 4
63, 5bitr4i 243 . . 3
7 2euswap 2232 . . . 4
8 df-reu 2563 . . . . 5
9 r19.42v 2707 . . . . . . . 8
10 df-rex 2562 . . . . . . . 8
119, 10bitr3i 242 . . . . . . 7
12 an12 772 . . . . . . . 8
1312exbii 1572 . . . . . . 7
1411, 13bitri 240 . . . . . 6
1514eubii 2165 . . . . 5
168, 15bitri 240 . . . 4
17 df-reu 2563 . . . . 5
18 r19.42v 2707 . . . . . . 7
19 df-rex 2562 . . . . . . 7
2018, 19bitr3i 242 . . . . . 6
2120eubii 2165 . . . . 5
2217, 21bitri 240 . . . 4
237, 16, 223imtr4g 261 . . 3
246, 23sylbi 187 . 2
252, 24sylbi 187 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wal 1530  wex 1531   wcel 1696  weu 2156  wmo 2157  wral 2556  wrex 2557  wreu 2558  wrmo 2559 This theorem is referenced by:  reuxfr2d  4573  reuxfr3d  23154 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564
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