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Theorem morex 3020
 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 2620 . . . 4 (x A φx(x A φ))
2 exancom 1586 . . . 4 (x(x A φ) ↔ x(φ x A))
31, 2bitri 240 . . 3 (x A φx(φ x A))
4 nfmo1 2215 . . . . . 6 x∃*xφ
5 nfe1 1732 . . . . . 6 xx(φ x A)
64, 5nfan 1824 . . . . 5 x(∃*xφ x(φ x A))
7 mopick 2266 . . . . 5 ((∃*xφ x(φ x A)) → (φx A))
86, 7alrimi 1765 . . . 4 ((∃*xφ x(φ x A)) → x(φx A))
9 morex.1 . . . . 5 B V
10 morex.2 . . . . . 6 (x = B → (φψ))
11 eleq1 2413 . . . . . 6 (x = B → (x AB A))
1210, 11imbi12d 311 . . . . 5 (x = B → ((φx A) ↔ (ψB A)))
139, 12spcv 2945 . . . 4 (x(φx A) → (ψB A))
148, 13syl 15 . . 3 ((∃*xφ x(φ x A)) → (ψB A))
153, 14sylan2b 461 . 2 ((∃*xφ x A φ) → (ψB A))
1615ancoms 439 1 ((x A φ ∃*xφ) → (ψB A))
 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  Vcvv 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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