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Theorem morex 2698
Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
morex.1 B V
morex.2 (x = B → (φψ))
Assertion
Ref Expression
morex ((x A φ ∃*xφ) → (ψB A))
Distinct variable groups:   x,B   x,A   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem morex
StepHypRef Expression
1 df-rex 2286 . . . 4 (x A φx(x A φ))
2 exancom 1477 . . . 4 (x(x A φ) ↔ x(φ x A))
31, 2bitri 173 . . 3 (x A φx(φ x A))
4 nfmo1 1890 . . . . . 6 x∃*xφ
5 nfe1 1362 . . . . . 6 xx(φ x A)
64, 5nfan 1435 . . . . 5 x(∃*xφ x(φ x A))
7 mopick 1956 . . . . 5 ((∃*xφ x(φ x A)) → (φx A))
86, 7alrimi 1392 . . . 4 ((∃*xφ x(φ x A)) → x(φx A))
9 morex.1 . . . . 5 B V
10 morex.2 . . . . . 6 (x = B → (φψ))
11 eleq1 2078 . . . . . 6 (x = B → (x AB A))
1210, 11imbi12d 223 . . . . 5 (x = B → ((φx A) ↔ (ψB A)))
139, 12spcv 2619 . . . 4 (x(φx A) → (ψB A))
148, 13syl 14 . . 3 ((∃*xφ x(φ x A)) → (ψB A))
153, 14sylan2b 271 . 2 ((∃*xφ x A φ) → (ψB A))
1615ancoms 255 1 ((x A φ ∃*xφ) → (ψB A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224   = wceq 1226  wex 1358   wcel 1370  ∃*wmo 1879  wrex 2281  Vcvv 2531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533
This theorem is referenced by: (None)
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