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Theorem bj-issetw 36791
Description: The closest one can get to isset 3497 without using ax-ext 2705. See also vexw 2717. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3497 using eleq2i 2830 (which requires ax-ext 2705 and df-cleq 2726). (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 36790 . 2 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
2 bj-issetw.1 . 2 𝜑
31, 2mpg 1795 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wex 1777  wcel 2103  {cab 2711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-clel 2813
This theorem is referenced by: (None)
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