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Theorem morex 3020
 Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
morex.1
morex.2
Assertion
Ref Expression
morex
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem morex
StepHypRef Expression
1 df-rex 2620 . . . 4
2 exancom 1586 . . . 4
31, 2bitri 240 . . 3
4 nfmo1 2215 . . . . . 6
5 nfe1 1732 . . . . . 6
64, 5nfan 1824 . . . . 5
7 mopick 2266 . . . . 5
86, 7alrimi 1765 . . . 4
9 morex.1 . . . . 5
10 morex.2 . . . . . 6
11 eleq1 2413 . . . . . 6
1210, 11imbi12d 311 . . . . 5
139, 12spcv 2945 . . . 4
148, 13syl 15 . . 3
153, 14sylan2b 461 . 2
1615ancoms 439 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wex 1541   wceq 1642   wcel 1710  wmo 2205  wrex 2615  cvv 2859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861 This theorem is referenced by: (None)
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