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Theorem morex 3283
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:   ,   ,   ,

Proof of Theorem morex
StepHypRef Expression
1 df-rex 2813 . . . 4
2 exancom 1671 . . . 4
31, 2bitri 249 . . 3
4 nfmo1 2295 . . . . . 6
5 nfe1 1840 . . . . . 6
64, 5nfan 1928 . . . . 5
7 mopick 2356 . . . . 5
86, 7alrimi 1877 . . . 4
9 morex.1 . . . . 5
10 morex.2 . . . . . 6
11 eleq1 2529 . . . . . 6
1210, 11imbi12d 320 . . . . 5
139, 12spcv 3200 . . . 4
148, 13syl 16 . . 3
153, 14sylan2b 475 . 2
1615ancoms 453 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  E*wmo 2283  E.wrex 2808   cvv 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-v 3111
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