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Theorem reupick2 3283
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reupick2 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reupick2
StepHypRef Expression
1 ancr 314 . . . . . 6 ((𝜓𝜑) → (𝜓 → (𝜑𝜓)))
21ralimi 2438 . . . . 5 (∀𝑥𝐴 (𝜓𝜑) → ∀𝑥𝐴 (𝜓 → (𝜑𝜓)))
3 rexim 2467 . . . . 5 (∀𝑥𝐴 (𝜓 → (𝜑𝜓)) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
42, 3syl 14 . . . 4 (∀𝑥𝐴 (𝜓𝜑) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
5 reupick3 3282 . . . . . 6 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
653exp 1142 . . . . 5 (∃!𝑥𝐴 𝜑 → (∃𝑥𝐴 (𝜑𝜓) → (𝑥𝐴 → (𝜑𝜓))))
76com12 30 . . . 4 (∃𝑥𝐴 (𝜑𝜓) → (∃!𝑥𝐴 𝜑 → (𝑥𝐴 → (𝜑𝜓))))
84, 7syl6 33 . . 3 (∀𝑥𝐴 (𝜓𝜑) → (∃𝑥𝐴 𝜓 → (∃!𝑥𝐴 𝜑 → (𝑥𝐴 → (𝜑𝜓)))))
983imp1 1156 . 2 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))
10 rsp 2423 . . . 4 (∀𝑥𝐴 (𝜓𝜑) → (𝑥𝐴 → (𝜓𝜑)))
11103ad2ant1 964 . . 3 ((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) → (𝑥𝐴 → (𝜓𝜑)))
1211imp 122 . 2 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜓𝜑))
139, 12impbid 127 1 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924  wcel 1438  wral 2359  wrex 2360  ∃!wreu 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-ral 2364  df-rex 2365  df-reu 2366
This theorem is referenced by: (None)
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