Theorem rabsneu 3696

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 2484	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
2	1	eqeq1i 2204	. . 3 ⊢ ({𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴} ↔ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} = {𝐴})
3		absneu 3695	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} = {𝐴}) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
4	2, 3	sylan2b 287	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
5		df-reu 2482	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
6	4, 5	sylibr 134	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃!weu 2045   ∈ wcel 2167  {cab 2182  ∃!wreu 2477  {crab 2479  {csn 3623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-reu 2482  df-rab 2484  df-v 2765  df-sn 3629

This theorem is referenced by: (None)