New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  reupick2 GIF version

Theorem reupick2 3541
 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 (((x A (ψφ) x A ψ ∃!x A φ) x A) → (φψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem reupick2
StepHypRef Expression
1 ancr 532 . . . . . 6 ((ψφ) → (ψ → (φ ψ)))
21ralimi 2689 . . . . 5 (x A (ψφ) → x A (ψ → (φ ψ)))
3 rexim 2718 . . . . 5 (x A (ψ → (φ ψ)) → (x A ψx A (φ ψ)))
42, 3syl 15 . . . 4 (x A (ψφ) → (x A ψx A (φ ψ)))
5 reupick3 3540 . . . . . 6 ((∃!x A φ x A (φ ψ) x A) → (φψ))
653exp 1150 . . . . 5 (∃!x A φ → (x A (φ ψ) → (x A → (φψ))))
76com12 27 . . . 4 (x A (φ ψ) → (∃!x A φ → (x A → (φψ))))
84, 7syl6 29 . . 3 (x A (ψφ) → (x A ψ → (∃!x A φ → (x A → (φψ)))))
983imp1 1164 . 2 (((x A (ψφ) x A ψ ∃!x A φ) x A) → (φψ))
10 rsp 2674 . . . 4 (x A (ψφ) → (x A → (ψφ)))
11103ad2ant1 976 . . 3 ((x A (ψφ) x A ψ ∃!x A φ) → (x A → (ψφ)))
1211imp 418 . 2 (((x A (ψφ) x A ψ ∃!x A φ) x A) → (ψφ))
139, 12impbid 183 1 (((x A (ψφ) x A ψ ∃!x A φ) x A) → (φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2619  df-rex 2620  df-reu 2621 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator