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Theorem euor 2062
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 1664 . . 3 𝜑 → ∀𝑥 ¬ 𝜑)
3 biorf 745 . . 3 𝜑 → (𝜓 ↔ (𝜑𝜓)))
42, 3eubidh 2042 . 2 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
54biimpa 296 1 ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  wal 1361  ∃!weu 2036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-eu 2039
This theorem is referenced by:  euorv  2063
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