Theorem euor 2636

Description: Introduce a disjunct into a unique existential quantifier. For a version requiring disjoint variables, but fewer axioms, see euorv 2637. (Contributed by NM, 21-Oct-2005.)

Hypothesis

Ref	Expression
euor.nf	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
euor	⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))

Proof of Theorem euor

Step	Hyp	Ref	Expression
1		euor.nf	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nfn 1889	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
3		biorf 950	. . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
4	2, 3	eubid 2612	. 2 ⊢ (¬ 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∨ 𝜓)))
5	4	biimpa 482	1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861  Ⅎwnf 1815  ∃!weu 2593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-mo 2564  df-eu 2594

This theorem is used by: (None)