Theorem intexabim 4242

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 2219	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2	1	exbii 1653	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
3		nfsab1 2221	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4		nfv 1576	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∣ 𝜑}
5		eleq1 2294	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
6	3, 4, 5	cbvex 1804	. . 3 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑})
7		inteximm 4239	. . 3 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
8	6, 7	sylbir 135	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
9	2, 8	sylbir 135	1 ⊢ (∃𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4  ∃wex 1540   ∈ wcel 2202  {cab 2217  Vcvv 2802  ∩ cint 3928

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-int 3929

This theorem is referenced by:  intexrabim  4243  omex  4691