Theorem euor2 2033

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.)

Assertion

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

Proof of Theorem euor2

Step	Hyp	Ref	Expression
1		hbe1 1454	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2	1	hbn 1615	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)
3		19.8a 1552	. . . 4 ⊢ (𝜑 → ∃𝑥𝜑)
4	3	con3i 604	. . 3 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
5		orel1 697	. . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓))
6		olc 683	. . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
7	5, 6	impbid1 141	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
8	4, 7	syl 14	. 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
9	2, 8	eubidh 1981	1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 680  ∃wex 1451  ∃!weu 1975

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-eu 1978

This theorem is referenced by:  reuun2  3325