Theorem intexabim 4215

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 2197	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2	1	exbii 1631	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
3		nfsab1 2199	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4		nfv 1554	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑥 ∣ 𝜑}
5		eleq1 2272	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
6	3, 4, 5	cbvex 1782	. . 3 ⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑})
7		inteximm 4212	. . 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 1518   ∈ wcel 2180  {cab 2195  Vcvv 2779  ∩ cint 3902

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191  ax-sep 4181

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-in 3183  df-ss 3190  df-int 3903

This theorem is referenced by:  intexrabim  4216  omex  4662