Theorem abn0m 3419

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 1508	. . 3 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∣ 𝜑}
2		nfsab1 2147	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
3		eleq1w 2218	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
4	1, 2, 3	cbvex 1736	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑})
5		abid 2145	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6	5	exbii 1585	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
7	4, 6	bitr3i 185	1 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 104  ∃wex 1472   ∈ wcel 2128  {cab 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-clel 2153

This theorem is referenced by:  mapprc  6598