ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  euor2 Structured version   GIF version

Theorem euor2 1955
Description: Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
euor2 xφ → (∃!x(φ ψ) ↔ ∃!xψ))

Proof of Theorem euor2
StepHypRef Expression
1 hbe1 1381 . . 3 (xφxxφ)
21hbn 1541 . 2 xφx ¬ xφ)
3 19.8a 1479 . . . 4 (φxφ)
43con3i 561 . . 3 xφ → ¬ φ)
5 orel1 643 . . . 4 φ → ((φ ψ) → ψ))
6 olc 631 . . . 4 (ψ → (φ ψ))
75, 6impbid1 130 . . 3 φ → ((φ ψ) ↔ ψ))
84, 7syl 14 . 2 xφ → ((φ ψ) ↔ ψ))
92, 8eubidh 1903 1 xφ → (∃!x(φ ψ) ↔ ∃!xψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 628  wex 1378  ∃!weu 1897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-eu 1900
This theorem is referenced by:  reuun2  3214
  Copyright terms: Public domain W3C validator