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Theorem bj-issetw 32054
 Description: The closest one can get to isset 3180 without using ax-ext 2590. See also bj-vexw 32049. Note that the only dv condition is between 𝑦 and 𝐴. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-issetw.1 𝜑
Assertion
Ref Expression
bj-issetw (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴)
Distinct variable group:   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem bj-issetw
StepHypRef Expression
1 bj-issetwt 32053 . 2 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
2 bj-issetw.1 . 2 𝜑
31, 2mpg 1715 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-clel 2606 This theorem is referenced by: (None)
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