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Theorem 2reu5 3110
 Description: Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2346 and reu3 3092. (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 2815 . . . . . . . 8
2 r19.29r 2815 . . . . . . . . 9
32reximi 2781 . . . . . . . 8
4 pm3.35 571 . . . . . . . . . . 11
54reximi 2781 . . . . . . . . . 10
65reximi 2781 . . . . . . . . 9
7 eleq1 2472 . . . . . . . . . . . . . 14
8 eleq1 2472 . . . . . . . . . . . . . 14
97, 8bi2anan9 844 . . . . . . . . . . . . 13
109biimpac 473 . . . . . . . . . . . 12
1110ancomd 439 . . . . . . . . . . 11
1211ex 424 . . . . . . . . . 10
1312rexlimivv 2803 . . . . . . . . 9
146, 13syl 16 . . . . . . . 8
151, 3, 143syl 19 . . . . . . 7
1615ex 424 . . . . . 6
1716pm4.71rd 617 . . . . 5
18 anass 631 . . . . 5
1917, 18syl6bb 253 . . . 4
20192exbidv 1635 . . 3
2120pm5.32i 619 . 2
22 2reu5lem3 3109 . 2
23 df-rex 2680 . . . 4
24 r19.42v 2830 . . . . . 6
25 df-rex 2680 . . . . . 6
2624, 25bitr3i 243 . . . . 5
2726exbii 1589 . . . 4
2823, 27bitri 241 . . 3
2928anbi2i 676 . 2
3021, 22, 293bitr4i 269 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1547   wcel 1721  wral 2674  wrex 2675  wreu 2676  wrmo 2677 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-cleq 2405  df-clel 2408  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682
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