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

Theorem iotauni 4979
Description: Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iotauni (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Proof of Theorem iotauni
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-eu 1951 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 iotaval 4978 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
3 uniabio 4977 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → {𝑥𝜑} = 𝑧)
42, 3eqtr4d 2123 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
54exlimiv 1534 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
61, 5sylbi 119 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wex 1426  ∃!weu 1948  {cab 2074   cuni 3648  cio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967
This theorem is referenced by:  iotaint  4980  fveu  5281  riotauni  5596
  Copyright terms: Public domain W3C validator