Theorem reusn 4656

Description: A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
reusn	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem reusn

Step	Hyp	Ref	Expression
1		euabsn2 4654	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
2		df-reu 3145	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rab 3147	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	eqeq1i 2826	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
5	4	exbii 1844	. 2 ⊢ (∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
6	1, 2, 5	3bitr4i 305	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃!weu 2649  {cab 2799  ∃!wreu 3140  {crab 3142  {csn 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-reu 3145  df-rab 3147  df-sn 4561

This theorem is referenced by:  reuen1  8572  cshwrepswhash1  16430  frcond3  28042  vdgn1frgrv2  28069  ddemeas  31490