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

Theorem absneu 3653
Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
Assertion
Ref Expression
absneu ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)

Proof of Theorem absneu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sneq 3592 . . . . 5 (𝑦 = 𝐴 → {𝑦} = {𝐴})
21eqeq2d 2182 . . . 4 (𝑦 = 𝐴 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝐴}))
32spcegv 2818 . . 3 (𝐴𝑉 → ({𝑥𝜑} = {𝐴} → ∃𝑦{𝑥𝜑} = {𝑦}))
43imp 123 . 2 ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃𝑦{𝑥𝜑} = {𝑦})
5 euabsn2 3650 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
64, 5sylibr 133 1 ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wex 1485  ∃!weu 2019  wcel 2141  {cab 2156  {csn 3581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sn 3587
This theorem is referenced by:  rabsneu  3654
  Copyright terms: Public domain W3C validator