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Theorem intexabim 4009
 Description: The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexabim (∃𝑥𝜑 {𝑥𝜑} ∈ V)

Proof of Theorem intexabim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 abid 2083 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21exbii 1548 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
3 nfsab1 2085 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1473 . . . 4 𝑦 𝑥 ∈ {𝑥𝜑}
5 eleq1 2157 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
63, 4, 5cbvex 1693 . . 3 (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥𝜑})
7 inteximm 4006 . . 3 (∃𝑦 𝑦 ∈ {𝑥𝜑} → {𝑥𝜑} ∈ V)
86, 7sylbir 134 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} → {𝑥𝜑} ∈ V)
92, 8sylbir 134 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∃wex 1433   ∈ wcel 1445  {cab 2081  Vcvv 2633  ∩ cint 3710 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-int 3711 This theorem is referenced by:  intexrabim  4010  omex  4436
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