Theorem rabsneu 4208

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 2905	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
2	1	eqeq1i 2615	. . 3 ⊢ ({𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴} ↔ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} = {𝐴})
3		absneu 4207	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} = {𝐴}) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
4	2, 3	sylan2b 491	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
5		df-reu 2903	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
6	4, 5	sylibr 223	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  {cab 2596  ∃!wreu 2898  {crab 2900  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-reu 2903  df-rab 2905  df-v 3175  df-sn 4126

This theorem is referenced by: (None)