Theorem clabel 2778

Description: Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
clabel	⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)))

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

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

Proof of Theorem clabel

Step	Hyp	Ref	Expression
1		df-clel 2647	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴))
2		abeq2 2761	. . . 4 ⊢ (𝑦 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
3	2	anbi2ci 732	. . 3 ⊢ ((𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)))
4	3	exbii 1814	. 2 ⊢ (∃𝑦(𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)))
5	1, 4	bitri 264	1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)))

Syntax hints:   ↔ wb 196   ∧ wa 383  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647

This theorem is referenced by:  sbabel  2822  grothprimlem  9693  ntrneiel2  38701