Theorem bj-abex 37085

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 3452	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
2		eqabb 2872	. . 3 ⊢ (𝑦 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
3	2	exbii 1849	. 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 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2711  Vcvv 3438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3440

This theorem is referenced by: (None)