Theorem rabeqsn 4161

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

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 2905	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)}
2		df-sn 4126	. . 3 ⊢ {𝑋} = {𝑥 ∣ 𝑥 = 𝑋}
3	1, 2	eqeq12i 2624	. 2 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑥 ∣ 𝑥 = 𝑋})
4		abbi 2724	. 2 ⊢ (∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋) ↔ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑥 ∣ 𝑥 = 𝑋})
5	3, 4	bitr4i 266	1 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋))

Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rab 2905  df-sn 4126

This theorem is referenced by:  k0004val0  37472  umgr2v2enb1  40742