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Theorem morex 3725
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 3071 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 exancom 1861 . . . 4 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥(𝜑𝑥𝐴))
31, 2bitri 275 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝜑𝑥𝐴))
4 nfmo1 2557 . . . . . 6 𝑥∃*𝑥𝜑
5 nfe1 2150 . . . . . 6 𝑥𝑥(𝜑𝑥𝐴)
64, 5nfan 1899 . . . . 5 𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴))
7 mopick 2625 . . . . 5 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜑𝑥𝐴))
86, 7alrimi 2213 . . . 4 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → ∀𝑥(𝜑𝑥𝐴))
9 morex.1 . . . . 5 𝐵 ∈ V
10 morex.2 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜓))
11 eleq1 2829 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
1210, 11imbi12d 344 . . . . 5 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜓𝐵𝐴)))
139, 12spcv 3605 . . . 4 (∀𝑥(𝜑𝑥𝐴) → (𝜓𝐵𝐴))
148, 13syl 17 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜓𝐵𝐴))
153, 14sylan2b 594 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥𝐴 𝜑) → (𝜓𝐵𝐴))
1615ancoms 458 1 ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  ∃*wmo 2538  wrex 3070  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482
This theorem is referenced by: (None)
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