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Theorem morex 3658
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 3072 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 exancom 1868 . . . 4 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥(𝜑𝑥𝐴))
31, 2bitri 274 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝜑𝑥𝐴))
4 nfmo1 2559 . . . . . 6 𝑥∃*𝑥𝜑
5 nfe1 2151 . . . . . 6 𝑥𝑥(𝜑𝑥𝐴)
64, 5nfan 1906 . . . . 5 𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴))
7 mopick 2629 . . . . 5 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜑𝑥𝐴))
86, 7alrimi 2210 . . . 4 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → ∀𝑥(𝜑𝑥𝐴))
9 morex.1 . . . . 5 𝐵 ∈ V
10 morex.2 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜓))
11 eleq1 2828 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
1210, 11imbi12d 345 . . . . 5 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜓𝐵𝐴)))
139, 12spcv 3543 . . . 4 (∀𝑥(𝜑𝑥𝐴) → (𝜓𝐵𝐴))
148, 13syl 17 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜓𝐵𝐴))
153, 14sylan2b 594 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥𝐴 𝜑) → (𝜓𝐵𝐴))
1615ancoms 459 1 ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1540   = wceq 1542  wex 1786  wcel 2110  ∃*wmo 2540  wrex 3067  Vcvv 3431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-clab 2718  df-cleq 2732  df-clel 2818  df-rex 3072  df-v 3433
This theorem is referenced by: (None)
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