Theorem euor 2071

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(𝜑 ∨ 𝜓))

Proof of Theorem euor

Step	Hyp	Ref	Expression
1		euor.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	hbn 1668	. . 3 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
3		biorf 745	. . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
4	2, 3	eubidh 2051	. 2 ⊢ (¬ 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∨ 𝜓)))
5	4	biimpa 296	1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  ∀wal 1362  ∃!weu 2045

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-eu 2048

This theorem is referenced by:  euorv  2072