Theorem rabn0m 3549

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 2534	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		rabid 2727	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
3	2	exbii 1658	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		nfv 1581	. . 3 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}
5		df-rab 2537	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	5	eleq2i 2305	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
7		nfsab1 2228	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
8	6, 7	nfxfr 1527	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}
9		eleq1 2301	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
10	4, 8, 9	cbvex 1809	. 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 1545   ∈ wcel 2209  {cab 2224  ∃wrex 2529  {crab 2532

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-rex 2534  df-rab 2537

This theorem is referenced by:  exss  4365  cc4f  7628  cc4n  7630  nnwosdc  12797  ballotfilemiex  13225  lspf  14701  incistruhgr  16248