Theorem euor2 2512

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 2025	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	nfn 1782	. 2 ⊢ Ⅎ𝑥 ¬ ∃𝑥𝜑
3		19.8a 2050	. . . 4 ⊢ (𝜑 → ∃𝑥𝜑)
4	3	con3i 150	. . 3 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
5		biorf 420	. . . 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 2486	1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383  ∃wex 1702  ∃!weu 2468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1703  df-nf 1708  df-eu 2472

This theorem is referenced by:  reuun2  3902