Theorem rabeqsn 4618

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 3394	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)}
2	1	eqeq1i 2735	. 2 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑋})
3		absn 4594	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋))
4	2, 3	bitri 275	1 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2110  {cab 2708  {crab 3393  {csn 4574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2722  df-rab 3394  df-sn 4575

This theorem is referenced by:  rabeqsnd  4620  rabsn  4672  made0  27811  umgr2v2enb1  29498  clwwlknon1loop  30068  wlkl0  30337  zarclssn  33876  k0004val0  44166