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Theorem bj-abex 36377
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 ({𝑥𝜑} ∈ V ↔ ∃𝑦𝑥(𝑥𝑦𝜑))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-abex
StepHypRef Expression
1 isset 3486 . 2 ({𝑥𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝜑})
2 eqabb 2872 . . 3 (𝑦 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝑦𝜑))
32exbii 1849 . 2 (∃𝑦 𝑦 = {𝑥𝜑} ↔ ∃𝑦𝑥(𝑥𝑦𝜑))
41, 3bitri 275 1 ({𝑥𝜑} ∈ V ↔ ∃𝑦𝑥(𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1538   = wceq 1540  wex 1780  wcel 2105  {cab 2708  Vcvv 3473
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475
This theorem is referenced by: (None)
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