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Theorem reupick2 4244
 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 550 . . . . . 6 ((𝜓𝜑) → (𝜓 → (𝜑𝜓)))
21ralimi 3131 . . . . 5 (∀𝑥𝐴 (𝜓𝜑) → ∀𝑥𝐴 (𝜓 → (𝜑𝜓)))
3 rexim 3207 . . . . 5 (∀𝑥𝐴 (𝜓 → (𝜑𝜓)) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
42, 3syl 17 . . . 4 (∀𝑥𝐴 (𝜓𝜑) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
5 reupick3 4243 . . . . . 6 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
653exp 1116 . . . . 5 (∃!𝑥𝐴 𝜑 → (∃𝑥𝐴 (𝜑𝜓) → (𝑥𝐴 → (𝜑𝜓))))
76com12 32 . . . 4 (∃𝑥𝐴 (𝜑𝜓) → (∃!𝑥𝐴 𝜑 → (𝑥𝐴 → (𝜑𝜓))))
84, 7syl6 35 . . 3 (∀𝑥𝐴 (𝜓𝜑) → (∃𝑥𝐴 𝜓 → (∃!𝑥𝐴 𝜑 → (𝑥𝐴 → (𝜑𝜓)))))
983imp1 1344 . 2 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))
10 rsp 3173 . . . 4 (∀𝑥𝐴 (𝜓𝜑) → (𝑥𝐴 → (𝜓𝜑)))
11103ad2ant1 1130 . . 3 ((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) → (𝑥𝐴 → (𝜓𝜑)))
1211imp 410 . 2 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜓𝜑))
139, 12impbid 215 1 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110  ∃!wreu 3111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-ral 3114  df-rex 3115  df-reu 3116 This theorem is referenced by:  grpoidval  28299  grpoidinv2  28301  grpoinv  28311
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