Theorem bj-abex 37206

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 3455	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
2		eqabb 2876	. . 3 ⊢ (𝑦 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
3	2	exbii 1850	. 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 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443

This theorem is referenced by: (None)