Theorem rabeqsn 4602

Description: Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 26-Aug-2022.)

Assertion

Ref	Expression
rabeqsn	⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋))

Distinct variable group:   𝑥,𝑋

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

Proof of Theorem rabeqsn

Step	Hyp	Ref	Expression
1		df-rab 3073	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)}
2	1	eqeq1i 2743	. 2 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑋})
3		absn 4579	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋))
4	2, 3	bitri 274	1 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  {crab 3068  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-rab 3073  df-sn 4562

This theorem is referenced by:  rabsn  4657  umgr2v2enb1  27893  clwwlknon1loop  28462  wlkl0  28731  rabeqsnd  30848  zarclssn  31823  made0  34057  k0004val0  41764