Theorem euor2 2647

Description: Introduce or eliminate a disjunct in a unique existential 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 2191	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	nfn 1884	. 2 ⊢ Ⅎ𝑥 ¬ ∃𝑥𝜑
3		19.8a 2223	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
4		biorf 949	. . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
5	4	bicomd 226	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
6	3, 5	nsyl5 160	. 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
7	2, 6	eubid 2621	1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860  ∃wex 1806  ∃!weu 2602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-mo 2573  df-eu 2603

This theorem is referenced by:  reuun2  4286