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Theorem bj-issetw 33375
 Description: The closest one can get to isset 3424 without using ax-ext 2803. See also bj-vexw 33369. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3424 using eleq2i 2898 (which requires ax-ext 2803 and df-cleq 2818). (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 33374 . 2 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
2 bj-issetw.1 . 2 𝜑
31, 2mpg 1896 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1656  ∃wex 1878   ∈ wcel 2164  {cab 2811 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-sb 2068  df-clab 2812  df-clel 2821 This theorem is referenced by: (None)
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