Theorem intexabim 4072

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 2125	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2	1	exbii 1584	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
3		nfsab1 2127	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4		nfv 1508	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∣ 𝜑}
5		eleq1 2200	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
6	3, 4, 5	cbvex 1729	. . 3 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑})
7		inteximm 4069	. . 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 1468   ∈ wcel 1480  {cab 2123  Vcvv 2681  ∩ cint 3766

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-int 3767

This theorem is referenced by:  intexrabim  4073  omex  4502