Theorem bj-issetw 36927

Description: The closest one can get to isset 3450 without using ax-ext 2703. See also vexw 2715. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3450 using eleq2i 2823 (which requires ax-ext 2703 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 ∈ {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 36926	. 2 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
2		bj-issetw.1	. 2 ⊢ 𝜑
3	1, 2	mpg 1798	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709

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 1911  ax-6 1968  ax-7 2009  ax-8 2113

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-clel 2806

This theorem is referenced by: (None)