Theorem reuun2 4325

Description: Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Wolf Lammen, 15-May-2025.)

Assertion

Ref	Expression
reuun2	⊢ (¬ ∃𝑥 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜑))

Proof of Theorem reuun2

Step	Hyp	Ref	Expression
1		df-rex 3071	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
2		euor2 2613	. . 3 ⊢ (¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → (∃!𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
3	1, 2	sylnbi 330	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐵 𝜑 → (∃!𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
4		df-reu 3381	. . 3 ⊢ (∃!𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃!𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑))
5		elun 4153	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
6	5	anbi1i 624	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝜑))
7		andir 1011	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)))
8		orcom 871	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)))
9	6, 7, 8	3bitri 297	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)))
10	9	eubii 2585	. . 3 ⊢ (∃!𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ∃!𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)))
11	4, 10	bitri 275	. 2 ⊢ (∃!𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃!𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)))
12		df-reu 3381	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
13	3, 11, 12	3bitr4g 314	1 ⊢ (¬ ∃𝑥 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848  ∃wex 1779   ∈ wcel 2108  ∃!weu 2568  ∃wrex 3070  ∃!wreu 3378   ∪ cun 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-reu 3381  df-v 3482  df-un 3956

This theorem is referenced by:  hdmap14lem4a  41873