Theorem euor 2052

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 1654	. . 3 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
3		biorf 744	. . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
4	2, 3	eubidh 2032	. 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 708  ∀wal 1351  ∃!weu 2026

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-eu 2029

This theorem is referenced by:  euorv  2053