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Theorem inrab 4205
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 3063 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 3063 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2ineq12i 4111 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)})
4 df-rab 3063 . . 3 {𝑥𝐴 ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
5 inab 4199 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐴𝜓))}
6 anandi 676 . . . . 5 ((𝑥𝐴 ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) ∧ (𝑥𝐴𝜓)))
76abbii 2804 . . . 4 {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐴𝜓))}
85, 7eqtr4i 2765 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
94, 8eqtr4i 2765 . 2 {𝑥𝐴 ∣ (𝜑𝜓)} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)})
103, 9eqtr4i 2765 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1542  wcel 2114  {cab 2717  {crab 3058  cin 3852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3402  df-in 3860
This theorem is referenced by:  rabnc  4286  ixxin  12851  hashbclem  13915  phiprmpw  16226  submacs  18120  ablfacrp  19320  dfrhm2  19604  ordtbaslem  21952  ordtbas2  21955  ordtopn3  21960  ordtcld3  21963  ordthauslem  22147  pthaus  22402  xkohaus  22417  tsmsfbas  22892  minveclem3b  24193  shftmbl  24303  mumul  25931  ppiub  25953  lgsquadlem2  26130  umgrislfupgrlem  27080  numedglnl  27102  clwwlknondisj  28061  frcond3  28219  numclwwlk3lem2  28334  xppreima  30570  xpinpreima  31441  xpinpreima2  31442  measvuni  31765  subfacp1lem6  32731  satfv1  32909  cnambfre  35481  itg2addnclem2  35485  ftc1anclem6  35511  refsymrels2  36335  dfeqvrels2  36357  refrelsredund4  36401  anrabdioph  40215  undisjrab  41503  smfaddlem2  43879  smfmullem4  43908
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