Theorem rabsneu 4688

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 3402	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
2	1	eqeq1i 2742	. . 3 ⊢ ({𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴} ↔ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} = {𝐴})
3		absneu 4687	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} = {𝐴}) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
4	2, 3	sylan2b 595	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
5		df-reu 3353	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
6	4, 5	sylibr 234	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃!weu 2569  {cab 2715  ∃!wreu 3350  {crab 3401  {csn 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-reu 3353  df-rab 3402  df-sn 4583

This theorem is referenced by: (None)