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Theorem intexrab 4954
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 4953 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rex 3067 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
3 df-rab 3070 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43inteqi 4615 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
54eleq1i 2841 . 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 382  wex 1852  wcel 2145  {cab 2757  wrex 3062  {crab 3065  Vcvv 3351   cint 4611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-int 4612
This theorem is referenced by:  onintrab2  7149  rankf  8821  rankvalb  8824  cardf2  8969  tskmval  9863  lspval  19188  aspval  19543  clsval  21062  spanval  28532  rgspnval  38264
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