Theorem euor2 2662

Description: Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)

Assertion

Ref	Expression
euor2	⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))

Proof of Theorem euor2

Step	Hyp	Ref	Expression
1		nfe1 2182	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	nfn 1934	. 2 ⊢ Ⅎ𝑥 ¬ ∃𝑥𝜑
3		19.8a 2205	. . . 4 ⊢ (𝜑 → ∃𝑥𝜑)
4	3	con3i 151	. . 3 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
5		biorf 896	. . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
6	5	bicomd 213	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
7	4, 6	syl 17	. 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
8	2, 7	eubid 2635	1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 826  ∃wex 1851  ∃!weu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-12 2202

This theorem depends on definitions:  df-bi 197  df-or 827  df-ex 1852  df-nf 1857  df-eu 2621

This theorem is referenced by:  reuun2  4056