Theorem abn0m 3463

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 1539	. . 3 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∣ 𝜑}
2		nfsab1 2179	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
3		eleq1w 2250	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
4	1, 2, 3	cbvex 1767	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
5		abid 2177	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6	5	exbii 1616	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
7	4, 6	bitr3i 186	1 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 105  ∃wex 1503   ∈ wcel 2160  {cab 2175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-clel 2185

This theorem is referenced by:  mapprc  6670