MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reupick2 Structured version   Visualization version   GIF version

Theorem reupick2 4321
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 548 . . . . . 6 ((𝜓𝜑) → (𝜓 → (𝜑𝜓)))
21ralimi 3084 . . . . 5 (∀𝑥𝐴 (𝜓𝜑) → ∀𝑥𝐴 (𝜓 → (𝜑𝜓)))
3 rexim 3088 . . . . 5 (∀𝑥𝐴 (𝜓 → (𝜑𝜓)) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
42, 3syl 17 . . . 4 (∀𝑥𝐴 (𝜓𝜑) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
5 reupick3 4320 . . . . . 6 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
653exp 1120 . . . . 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 1348 . 2 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))
10 rsp 3245 . . . 4 (∀𝑥𝐴 (𝜓𝜑) → (𝑥𝐴 → (𝜓𝜑)))
11103ad2ant1 1134 . . 3 ((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) → (𝑥𝐴 → (𝜓𝜑)))
1211imp 408 . 2 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜓𝜑))
139, 12impbid 211 1 (((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wcel 2107  wral 3062  wrex 3071  ∃!wreu 3375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-ex 1783  df-nf 1787  df-mo 2535  df-eu 2564  df-ral 3063  df-rex 3072  df-reu 3378
This theorem is referenced by:  grpoidval  29766  grpoidinv2  29768  grpoinv  29778
  Copyright terms: Public domain W3C validator