Theorem abn0m 3327

Description: Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)

Assertion

Ref	Expression
abn0m	⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abn0m

Step	Hyp	Ref	Expression
1		nfv 1473	. . 3 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∣ 𝜑}
2		nfsab1 2085	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
3		eleq1w 2155	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
4	1, 2, 3	cbvex 1693	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
5		abid 2083	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6	5	exbii 1548	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
7	4, 6	bitr3i 185	1 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 104  ∃wex 1433   ∈ wcel 1445  {cab 2081

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-clel 2091

This theorem is referenced by:  mapprc  6449