Theorem intexabim 3927

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

Assertion

Ref	Expression
intexabim	⊢ (∃𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} ∈ V)

Proof of Theorem intexabim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abid 2069	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2	1	exbii 1536	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
3		nfsab1 2071	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4		nfv 1461	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∣ 𝜑}
5		eleq1 2141	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
6	3, 4, 5	cbvex 1679	. . 3 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑})
7		inteximm 3924	. . 3 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
8	6, 7	sylbir 133	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
9	2, 8	sylbir 133	1 ⊢ (∃𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4  ∃wex 1421   ∈ wcel 1433  {cab 2067  Vcvv 2601  ∩ cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-int 3637

This theorem is referenced by:  intexrabim  3928  omex  4334