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Theorem bj-abex 35849
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 3488 . 2 ({𝑥𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝜑})
2 eqabb 2874 . . 3 (𝑦 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝑦𝜑))
32exbii 1851 . 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 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  Vcvv 3475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477
This theorem is referenced by: (None)
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