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 ∈ {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 33374	. 2 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
2		bj-issetw.1	. 2 ⊢ 𝜑
3	1, 2	mpg 1896	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)

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)