Theorem intexrabim 3898

Description: The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
intexrabim	⊢ (∃x ∈ A φ → ∩ {x ∈ A ∣ φ} ∈ V)

Proof of Theorem intexrabim

Step	Hyp	Ref	Expression
1		intexabim 3897	. 2 ⊢ (∃x(x ∈ A ∧ φ) → ∩ {x ∣ (x ∈ A ∧ φ)} ∈ V)
2		df-rex 2306	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
3		df-rab 2309	. . . 4 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
4	3	inteqi 3610	. . 3 ⊢ ∩ {x ∈ A ∣ φ} = ∩ {x ∣ (x ∈ A ∧ φ)}
5	4	eleq1i 2100	. 2 ⊢ (∩ {x ∈ A ∣ φ} ∈ V ↔ ∩ {x ∣ (x ∈ A ∧ φ)} ∈ V)
6	1, 2, 5	3imtr4i 190	1 ⊢ (∃x ∈ A φ → ∩ {x ∈ A ∣ φ} ∈ V)

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  {crab 2304  Vcvv 2551  ∩ cint 3606

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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-int 3607

This theorem is referenced by: (None)