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Theorem intexrab 4793
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 4792 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rex 2914 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
3 df-rab 2917 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43inteqi 4451 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
54eleq1i 2689 . 2 ( {𝑥𝐴𝜑} ∈ V ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
61, 2, 53bitr4i 292 1 (∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1701  wcel 1987  {cab 2607  wrex 2909  {crab 2912  Vcvv 3190   cint 4447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-in 3567  df-ss 3574  df-nul 3898  df-int 4448
This theorem is referenced by:  onintrab2  6964  rankf  8617  rankvalb  8620  cardf2  8729  tskmval  9621  lspval  18915  aspval  19268  clsval  20781  spanval  28080  rgspnval  37258
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