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Theorem 2reu1 27921
 Description: Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2360. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
2reu1
Distinct variable groups:   ,,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem 2reu1
StepHypRef Expression
1 2reu5a 27912 . . . . . . 7
21simprbi 451 . . . . . 6
3 simprr 734 . . . . . . . . . . . 12
4 rsp 2758 . . . . . . . . . . . . . 14
54adantr 452 . . . . . . . . . . . . 13
65impcom 420 . . . . . . . . . . . 12
73, 6jca 519 . . . . . . . . . . 11
87ex 424 . . . . . . . . . 10
98rmoimia 3126 . . . . . . . . 9
10 nfra1 2748 . . . . . . . . . 10
1110rmoanim 27914 . . . . . . . . 9
129, 11sylib 189 . . . . . . . 8
1312ancrd 538 . . . . . . 7
14 2rmoswap 27919 . . . . . . . . 9
1514com12 29 . . . . . . . 8
1615imdistani 672 . . . . . . 7
1713, 16syl6 31 . . . . . 6
182, 17syl 16 . . . . 5
19 2reu2rex 27918 . . . . . 6
20 rexcom 2861 . . . . . . 7
2119, 20sylib 189 . . . . . 6
2219, 21jca 519 . . . . 5
2318, 22jctild 528 . . . 4
24 reu5 2913 . . . . . 6
25 reu5 2913 . . . . . 6
2624, 25anbi12i 679 . . . . 5
27 an4 798 . . . . 5
2826, 27bitri 241 . . . 4
2923, 28syl6ibr 219 . . 3
3029com12 29 . 2
31 2rexreu 27920 . 2
3230, 31impbid1 195 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wcel 1725  wral 2697  wrex 2698  wreu 2699  wrmo 2700 This theorem is referenced by:  2reu2  27922  2reu3  27923 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  ax-ext 2416 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 2284  df-mo 2285  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705
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