Theorem reuun2 3539

Description: Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)

Assertion

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

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem reuun2

Step	Hyp	Ref	Expression
1		df-rex 2621	. . 3 ⊢ (∃x ∈ B φ ↔ ∃x(x ∈ B ∧ φ))
2		euor2 2272	. . 3 ⊢ (¬ ∃x(x ∈ B ∧ φ) → (∃!x((x ∈ B ∧ φ) ∨ (x ∈ A ∧ φ)) ↔ ∃!x(x ∈ A ∧ φ)))
3	1, 2	sylnbi 297	. 2 ⊢ (¬ ∃x ∈ B φ → (∃!x((x ∈ B ∧ φ) ∨ (x ∈ A ∧ φ)) ↔ ∃!x(x ∈ A ∧ φ)))
4		df-reu 2622	. . 3 ⊢ (∃!x ∈ (A ∪ B)φ ↔ ∃!x(x ∈ (A ∪ B) ∧ φ))
5		elun 3221	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
6	5	anbi1i 676	. . . . 5 ⊢ ((x ∈ (A ∪ B) ∧ φ) ↔ ((x ∈ A ∨ x ∈ B) ∧ φ))
7		andir 838	. . . . . 6 ⊢ (((x ∈ A ∨ x ∈ B) ∧ φ) ↔ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ φ)))
8		orcom 376	. . . . . 6 ⊢ (((x ∈ A ∧ φ) ∨ (x ∈ B ∧ φ)) ↔ ((x ∈ B ∧ φ) ∨ (x ∈ A ∧ φ)))
9	7, 8	bitri 240	. . . . 5 ⊢ (((x ∈ A ∨ x ∈ B) ∧ φ) ↔ ((x ∈ B ∧ φ) ∨ (x ∈ A ∧ φ)))
10	6, 9	bitri 240	. . . 4 ⊢ ((x ∈ (A ∪ B) ∧ φ) ↔ ((x ∈ B ∧ φ) ∨ (x ∈ A ∧ φ)))
11	10	eubii 2213	. . 3 ⊢ (∃!x(x ∈ (A ∪ B) ∧ φ) ↔ ∃!x((x ∈ B ∧ φ) ∨ (x ∈ A ∧ φ)))
12	4, 11	bitri 240	. 2 ⊢ (∃!x ∈ (A ∪ B)φ ↔ ∃!x((x ∈ B ∧ φ) ∨ (x ∈ A ∧ φ)))
13		df-reu 2622	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
14	3, 12, 13	3bitr4g 279	1 ⊢ (¬ ∃x ∈ B φ → (∃!x ∈ (A ∪ B)φ ↔ ∃!x ∈ A φ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃wrex 2616  ∃!wreu 2617   ∪ cun 3208

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-reu 2622  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215

This theorem is used by: (None)