Theorem bj-issetw 37539

Description: The closest one can get to isset 3468 without using ax-ext 2734. See also vexw 2746. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3468 using eleq2i 2854 (which requires ax-ext 2734 and df-cleq 2754). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-issetw.1	⊢ 𝜑

Assertion

Ref	Expression
bj-issetw	⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)

Distinct variable group:   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥)

Proof of Theorem bj-issetw

Step	Hyp	Ref	Expression
1		bj-issetwt 37538	. 2 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
2		bj-issetw.1	. 2 ⊢ 𝜑
3	1, 2	mpg 1826	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-clel 2837

This theorem is used by: (None)