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Theorem 2reurex 37567
 Description: Double restricted quantification with existential uniqueness, analogous to 2euex 2319. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
Assertion
Ref Expression
2reurex
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   (,)   ()   ()

Proof of Theorem 2reurex
StepHypRef Expression
1 reu5 3025 . 2
2 rexcom 2971 . . . 4
3 nfcv 2566 . . . . . 6
4 nfre1 2867 . . . . . 6
53, 4nfrmo 2985 . . . . 5
6 rspe 2864 . . . . . . . . . . 11
76ex 434 . . . . . . . . . 10
87ralrimivw 2821 . . . . . . . . 9
9 rmoim 3251 . . . . . . . . 9
108, 9syl 17 . . . . . . . 8
1110impcom 430 . . . . . . 7
12 rmo5 3028 . . . . . . 7
1311, 12sylib 198 . . . . . 6
1413ex 434 . . . . 5
155, 14reximdai 2875 . . . 4
162, 15syl5bi 219 . . 3
1716impcom 430 . 2
181, 17sylbi 197 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wcel 1844  wral 2756  wrex 2757  wreu 2758  wrmo 2759 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382 This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764 This theorem is referenced by:  2rexreu  37571
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