Theorem euor2 1958

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

Assertion

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

Proof of Theorem euor2

Step	Hyp	Ref	Expression
1		hbe1 1384	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2	1	hbn 1544	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)
3		19.8a 1482	. . . 4 ⊢ (𝜑 → ∃𝑥𝜑)
4	3	con3i 562	. . 3 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
5		orel1 644	. . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓))
6		olc 632	. . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
7	5, 6	impbid1 130	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
8	4, 7	syl 14	. 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
9	2, 8	eubidh 1906	1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 629  ∃wex 1381  ∃!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-eu 1903

This theorem is referenced by:  reuun2  3220