New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  reupick3 GIF version

Theorem reupick3 3540
 Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reupick3 ((∃!x A φ x A (φ ψ) x A) → (φψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem reupick3
StepHypRef Expression
1 df-reu 2621 . . . 4 (∃!x A φ∃!x(x A φ))
2 df-rex 2620 . . . . 5 (x A (φ ψ) ↔ x(x A (φ ψ)))
3 anass 630 . . . . . 6 (((x A φ) ψ) ↔ (x A (φ ψ)))
43exbii 1582 . . . . 5 (x((x A φ) ψ) ↔ x(x A (φ ψ)))
52, 4bitr4i 243 . . . 4 (x A (φ ψ) ↔ x((x A φ) ψ))
6 eupick 2267 . . . 4 ((∃!x(x A φ) x((x A φ) ψ)) → ((x A φ) → ψ))
71, 5, 6syl2anb 465 . . 3 ((∃!x A φ x A (φ ψ)) → ((x A φ) → ψ))
87exp3a 425 . 2 ((∃!x A φ x A (φ ψ)) → (x A → (φψ)))
983impia 1148 1 ((∃!x A φ x A (φ ψ) x A) → (φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃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-rex 2620  df-reu 2621 This theorem is referenced by:  reupick2  3541
 Copyright terms: Public domain W3C validator