Theorem euor2 2100

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 1506	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2	1	hbn 1665	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)
3		19.8a 1601	. . . 4 ⊢ (𝜑 → ∃𝑥𝜑)
4	3	con3i 633	. . 3 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
5		orel1 726	. . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓))
6		olc 712	. . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
7	5, 6	impbid1 142	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
8	4, 7	syl 14	. 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
9	2, 8	eubidh 2048	1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 709  ∃wex 1503  ∃!weu 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-eu 2045

This theorem is referenced by:  reuun2  3442