Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-issetw Structured version   Visualization version   GIF version

Theorem bj-issetw 36219
Description: The closest one can get to isset 3486 without using ax-ext 2702. See also vexw 2714. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3486 using eleq2i 2824 (which requires ax-ext 2702 and df-cleq 2723). (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 36218 . 2 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
2 bj-issetw.1 . 2 𝜑
31, 2mpg 1798 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1540  wex 1780  wcel 2105  {cab 2708
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
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-clel 2809
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator