Theorem intexabim 4113

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 2145	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2	1	exbii 1585	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
3		nfsab1 2147	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4		nfv 1508	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∣ 𝜑}
5		eleq1 2220	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
6	3, 4, 5	cbvex 1736	. . 3 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑})
7		inteximm 4110	. . 3 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
8	6, 7	sylbir 134	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
9	2, 8	sylbir 134	1 ⊢ (∃𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4  ∃wex 1472   ∈ wcel 2128  {cab 2143  Vcvv 2712  ∩ cint 3807

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4082

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-int 3808

This theorem is referenced by:  intexrabim  4114  omex  4552