Theorem intexrabim 4182

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 4181	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
2		df-rex 2478	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rab 2481	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	inteqi 3874	. . 3 ⊢ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} = ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
5	4	eleq1i 2259	. 2 ⊢ (∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
6	1, 2, 5	3imtr4i 201	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1503   ∈ wcel 2164  {cab 2179  ∃wrex 2473  {crab 2476  Vcvv 2760  ∩ cint 3870

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sep 4147

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-in 3159  df-ss 3166  df-int 3871

This theorem is referenced by:  cardcl  7241  isnumi  7242  cardval3ex  7245  lspval  13886  clsval  14279