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Theorem morex 3110
 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 2703 . . . 4
2 exancom 1596 . . . 4
31, 2bitri 241 . . 3
4 nfmo1 2291 . . . . . 6
5 nfe1 1747 . . . . . 6
64, 5nfan 1846 . . . . 5
7 mopick 2342 . . . . 5
86, 7alrimi 1781 . . . 4
9 morex.1 . . . . 5
10 morex.2 . . . . . 6
11 eleq1 2495 . . . . . 6
1210, 11imbi12d 312 . . . . 5
139, 12spcv 3034 . . . 4
148, 13syl 16 . . 3
153, 14sylan2b 462 . 2
1615ancoms 440 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549  wex 1550   wceq 1652   wcel 1725  wmo 2281  wrex 2698  cvv 2948 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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950
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