Theorem rabn0m 3436

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 2450	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		rabid 2641	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
3	2	exbii 1593	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		nfv 1516	. . 3 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}
5		df-rab 2453	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	5	eleq2i 2233	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
7		nfsab1 2155	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
8	6, 7	nfxfr 1462	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}
9		eleq1 2229	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
10	4, 8, 9	cbvex 1744	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
11	1, 3, 10	3bitr2ri 208	1 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1480   ∈ wcel 2136  {cab 2151  ∃wrex 2445  {crab 2448

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-rex 2450  df-rab 2453

This theorem is referenced by:  exss  4205  cc4f  7210  cc4n  7212  nnwosdc  11972