Theorem bj-issetw 32860

Description: The closest one can get to isset 3207 without using ax-ext 2602. See also bj-vexw 32855. Note that the only dv condition is between 𝑦 and 𝐴. (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 32859	. 2 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
2		bj-issetw.1	. 2 ⊢ 𝜑
3	1, 2	mpg 1724	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)

Syntax hints:   ↔ wb 196   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-clel 2618

This theorem is referenced by: (None)