Theorem intexrabim 4139

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 4138	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
2		df-rex 2454	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rab 2457	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	inteqi 3835	. . 3 ⊢ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} = ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
5	4	eleq1i 2236	. 2 ⊢ (∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
6	1, 2, 5	3imtr4i 200	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1485   ∈ wcel 2141  {cab 2156  ∃wrex 2449  {crab 2452  Vcvv 2730  ∩ cint 3831

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-int 3832

This theorem is referenced by:  cardcl  7158  isnumi  7159  cardval3ex  7162  clsval  12905