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

Theorem modc 1925
 Description: Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
Hypothesis
Ref Expression
modc.1 yφ
Assertion
Ref Expression
modc (DECID xφ → (yx(φx = y) ↔ xy((φ [y / x]φ) → x = y)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem modc
StepHypRef Expression
1 modc.1 . . 3 yφ
21mo23 1923 . 2 (yx(φx = y) → xy((φ [y / x]φ) → x = y))
3 exmiddc 735 . . 3 (DECID xφ → (xφ ¬ xφ))
41mor 1924 . . . 4 (xφ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
51mo2n 1910 . . . . 5 xφyx(φx = y))
65a1d 22 . . . 4 xφ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
74, 6jaoi 623 . . 3 ((xφ ¬ xφ) → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
83, 7syl 14 . 2 (DECID xφ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
92, 8impbid2 131 1 (DECID xφ → (yx(φx = y) ↔ xy((φ [y / x]φ) → x = y)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616  DECID wdc 733  ∀wal 1226  Ⅎwnf 1329  ∃wex 1362  [wsb 1627 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410 This theorem depends on definitions:  df-bi 110  df-dc 734  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628 This theorem is referenced by:  mo2dc  1937
 Copyright terms: Public domain W3C validator