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Theorem morex 2962
 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 2562 . . . 4
2 exancom 1576 . . . 4
31, 2bitri 240 . . 3
4 nfmo1 2167 . . . . . 6
5 nfe1 1718 . . . . . 6
64, 5nfan 1783 . . . . 5
7 mopick 2218 . . . . 5
86, 7alrimi 1757 . . . 4
9 morex.1 . . . . 5
10 morex.2 . . . . . 6
11 eleq1 2356 . . . . . 6
1210, 11imbi12d 311 . . . . 5
139, 12spcv 2887 . . . 4
148, 13syl 15 . . 3
153, 14sylan2b 461 . 2
1615ancoms 439 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1530  wex 1531   wceq 1632   wcel 1696  wmo 2157  wrex 2557  cvv 2801 This theorem is referenced by:  morexOLD  26450 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803
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