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Theorem intexrab 5219
 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 5218 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rex 3131 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
3 df-rab 3134 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43inteqi 4856 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
54eleq1i 2901 . 2 ( {𝑥𝐴𝜑} ∈ V ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
61, 2, 53bitr4i 305 1 (∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1780   ∈ wcel 2114  {cab 2798  ∃wrex 3126  {crab 3129  Vcvv 3473  ∩ cint 4852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4270  df-int 4853 This theorem is referenced by:  onintrab2  7495  rankf  9201  rankvalb  9204  cardf2  9350  tskmval  10239  lspval  19723  aspval  20078  clsval  21621  spanval  29095  rgspnval  39905
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