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Theorem intexrab 5353
Description: The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
Assertion
Ref Expression
intexrab (∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)

Proof of Theorem intexrab
StepHypRef Expression
1 intexab 5352 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rex 3069 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
3 df-rab 3434 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43inteqi 4955 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
54eleq1i 2830 . 2 ( {𝑥𝐴𝜑} ∈ V ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
61, 2, 53bitr4i 303 1 (∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1776  wcel 2106  {cab 2712  wrex 3068  {crab 3433  Vcvv 3478   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-int 4952
This theorem is referenced by:  onintrab2  7817  rankf  9832  rankvalb  9835  cardf2  9981  tskmval  10877  rgspnval  20629  lspval  20991  aspval  21911  clsval  23061  spanval  31362  fldgenval  33294
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