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Theorem inrab 4240
Description: Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
Assertion
Ref Expression
inrab ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Proof of Theorem inrab
StepHypRef Expression
1 df-rab 3073 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 3073 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2ineq12i 4144 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)})
4 df-rab 3073 . . 3 {𝑥𝐴 ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
5 inab 4233 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐴𝜓))}
6 anandi 673 . . . . 5 ((𝑥𝐴 ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) ∧ (𝑥𝐴𝜓)))
76abbii 2808 . . . 4 {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐴𝜓))}
85, 7eqtr4i 2769 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
94, 8eqtr4i 2769 . 2 {𝑥𝐴 ∣ (𝜑𝜓)} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)})
103, 9eqtr4i 2769 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wcel 2106  {cab 2715  {crab 3068  cin 3886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894
This theorem is referenced by:  rabnc  4321  ixxin  13096  hashbclem  14164  phiprmpw  16477  submacs  18465  ablfacrp  19669  dfrhm2  19961  ordtbaslem  22339  ordtbas2  22342  ordtopn3  22347  ordtcld3  22350  ordthauslem  22534  pthaus  22789  xkohaus  22804  tsmsfbas  23279  minveclem3b  24592  shftmbl  24702  mumul  26330  ppiub  26352  lgsquadlem2  26529  umgrislfupgrlem  27492  numedglnl  27514  clwwlknondisj  28475  frcond3  28633  numclwwlk3lem2  28748  xppreima  30983  xpinpreima  31856  xpinpreima2  31857  measvuni  32182  subfacp1lem6  33147  satfv1  33325  cnambfre  35825  itg2addnclem2  35829  ftc1anclem6  35855  refsymrels2  36679  dfeqvrels2  36701  refrelsredund4  36745  anrabdioph  40602  undisjrab  41924  smfaddlem2  44299  smfmullem4  44328
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