Theorem clabel 2356

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 2225	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴))
2		abeq2 2338	. . . 4 ⊢ (𝑦 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
3	2	anbi2ci 459	. . 3 ⊢ ((𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)))
4	3	exbii 1651	. 2 ⊢ (∃𝑦(𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)))
5	1, 4	bitri 184	1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1393   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225

This theorem is referenced by:  frecabcl  6551