Theorem rabeqsn 4214

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 2921	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)}
2		df-sn 4178	. . 3 ⊢ {𝑋} = {𝑥 ∣ 𝑥 = 𝑋}
3	1, 2	eqeq12i 2636	. 2 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑥 ∣ 𝑥 = 𝑋})
4		abbi 2737	. 2 ⊢ (∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋) ↔ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑥 ∣ 𝑥 = 𝑋})
5	3, 4	bitr4i 267	1 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  {crab 2916  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921  df-sn 4178

This theorem is referenced by:  umgr2v2enb1  26422  rabeqsnd  29342  k0004val0  38452