Theorem bj-abex 36990

Description: Two ways of stating that the extension of a formula is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-abex	⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-abex

Step	Hyp	Ref	Expression
1		isset 3477	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
2		eqabb 2873	. . 3 ⊢ (𝑦 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
3	2	exbii 1847	. 2 ⊢ (∃𝑦 𝑦 = {𝑥 ∣ 𝜑} ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
4	1, 3	bitri 275	1 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))

Syntax hints:   ↔ wb 206  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2712  Vcvv 3463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465

This theorem is referenced by: (None)