Theorem bj-issetw 37366

Description: The closest one can get to isset 3469 without using ax-ext 2735. See also vexw 2747. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3469 using eleq2i 2855 (which requires ax-ext 2735 and df-cleq 2755). (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 37365	. 2 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
2		bj-issetw.1	. 2 ⊢ 𝜑
3	1, 2	mpg 1818	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)

Syntax hints:   ↔ wb 208   = wceq 1561  ∃wex 1800   ∈ wcel 2143  {cab 2741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-clel 2838

This theorem is referenced by: (None)