Theorem rabsneu 3796

Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
rabsneu	⊢ ((A ∈ V ∧ {x ∈ B ∣ φ} = {A}) → ∃!x ∈ B φ)

Proof of Theorem rabsneu

Step	Hyp	Ref	Expression
1		df-rab 2624	. . . 4 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)}
2	1	eqeq1i 2360	. . 3 ⊢ ({x ∈ B ∣ φ} = {A} ↔ {x ∣ (x ∈ B ∧ φ)} = {A})
3		absneu 3795	. . 3 ⊢ ((A ∈ V ∧ {x ∣ (x ∈ B ∧ φ)} = {A}) → ∃!x(x ∈ B ∧ φ))
4	2, 3	sylan2b 461	. 2 ⊢ ((A ∈ V ∧ {x ∈ B ∣ φ} = {A}) → ∃!x(x ∈ B ∧ φ))
5		df-reu 2622	. 2 ⊢ (∃!x ∈ B φ ↔ ∃!x(x ∈ B ∧ φ))
6	4, 5	sylibr 203	1 ⊢ ((A ∈ V ∧ {x ∈ B ∣ φ} = {A}) → ∃!x ∈ B φ)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  ∃!wreu 2617  {crab 2619  {csn 3738

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-reu 2622  df-rab 2624  df-v 2862  df-sn 3742

This theorem is referenced by: (None)