Theorem rabn0m 3239

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 2306	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
2		rabid 2479	. . 3 ⊢ (x ∈ {x ∈ A ∣ φ} ↔ (x ∈ A ∧ φ))
3	2	exbii 1493	. 2 ⊢ (∃x x ∈ {x ∈ A ∣ φ} ↔ ∃x(x ∈ A ∧ φ))
4		nfv 1418	. . 3 ⊢ Ⅎy x ∈ {x ∈ A ∣ φ}
5		df-rab 2309	. . . . 5 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
6	5	eleq2i 2101	. . . 4 ⊢ (y ∈ {x ∈ A ∣ φ} ↔ y ∈ {x ∣ (x ∈ A ∧ φ)})
7		nfsab1 2027	. . . 4 ⊢ Ⅎx y ∈ {x ∣ (x ∈ A ∧ φ)}
8	6, 7	nfxfr 1360	. . 3 ⊢ Ⅎx y ∈ {x ∈ A ∣ φ}
9		eleq1 2097	. . 3 ⊢ (x = y → (x ∈ {x ∈ A ∣ φ} ↔ y ∈ {x ∈ A ∣ φ}))
10	4, 8, 9	cbvex 1636	. 2 ⊢ (∃x x ∈ {x ∈ A ∣ φ} ↔ ∃y y ∈ {x ∈ A ∣ φ})
11	1, 3, 10	3bitr2ri 198	1 ⊢ (∃y y ∈ {x ∈ A ∣ φ} ↔ ∃x ∈ A φ)

Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  {crab 2304

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-rex 2306  df-rab 2309

This theorem is referenced by:  exss  3954