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Theorem morex 3713
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 ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem morex
StepHypRef Expression
1 df-rex 3061 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 exancom 1857 . . . 4 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥(𝜑𝑥𝐴))
31, 2bitri 274 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝜑𝑥𝐴))
4 nfmo1 2546 . . . . . 6 𝑥∃*𝑥𝜑
5 nfe1 2140 . . . . . 6 𝑥𝑥(𝜑𝑥𝐴)
64, 5nfan 1895 . . . . 5 𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴))
7 mopick 2614 . . . . 5 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜑𝑥𝐴))
86, 7alrimi 2202 . . . 4 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → ∀𝑥(𝜑𝑥𝐴))
9 morex.1 . . . . 5 𝐵 ∈ V
10 morex.2 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜓))
11 eleq1 2814 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
1210, 11imbi12d 343 . . . . 5 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜓𝐵𝐴)))
139, 12spcv 3591 . . . 4 (∀𝑥(𝜑𝑥𝐴) → (𝜓𝐵𝐴))
148, 13syl 17 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜓𝐵𝐴))
153, 14sylan2b 592 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥𝐴 𝜑) → (𝜓𝐵𝐴))
1615ancoms 457 1 ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1532   = wceq 1534  wex 1774  wcel 2099  ∃*wmo 2527  wrex 3060  Vcvv 3462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-clab 2704  df-cleq 2718  df-clel 2803  df-rex 3061  df-v 3464
This theorem is referenced by: (None)
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