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

Theorem hbeu 1970
Description: Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.)
Hypothesis
Ref Expression
hbeu.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbeu (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)

Proof of Theorem hbeu
StepHypRef Expression
1 hbeu.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21nfi 1397 . . 3 𝑥𝜑
32nfeu 1968 . 2 𝑥∃!𝑦𝜑
43nfri 1458 1 (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1288  ∃!weu 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952
This theorem is referenced by:  hbmo  1988  2eu7  2043
  Copyright terms: Public domain W3C validator