Theorem rabn0m 3519

Description: Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)

Assertion

Ref	Expression
rabn0m	⊢ (∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rabn0m

Step	Hyp	Ref	Expression
1		df-rex 2514	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		rabid 2707	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
3	2	exbii 1651	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		nfv 1574	. . 3 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}
5		df-rab 2517	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	5	eleq2i 2296	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
7		nfsab1 2219	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
8	6, 7	nfxfr 1520	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}
9		eleq1 2292	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
10	4, 8, 9	cbvex 1802	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
11	1, 3, 10	3bitr2ri 209	1 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∃wrex 2509  {crab 2512

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-rex 2514  df-rab 2517

This theorem is referenced by:  exss  4313  cc4f  7455  cc4n  7457  nnwosdc  12560  lspf  14353  incistruhgr  15890