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

Theorem euor 1971
Description: Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.)
Hypothesis
Ref Expression
euor.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
euor ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Proof of Theorem euor
StepHypRef Expression
1 euor.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21hbn 1587 . . 3 𝜑 → ∀𝑥 ¬ 𝜑)
3 biorf 696 . . 3 𝜑 → (𝜓 ↔ (𝜑𝜓)))
42, 3eubidh 1951 . 2 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
54biimpa 290 1 ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  wal 1285  ∃!weu 1945
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-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-eu 1948
This theorem is referenced by:  euorv  1972
  Copyright terms: Public domain W3C validator