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Theorem inrab 3318
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 2402 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 2402 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2ineq12i 3245 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)})
4 df-rab 2402 . . 3 {𝑥𝐴 ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
5 inab 3314 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐴𝜓))}
6 anandi 564 . . . . 5 ((𝑥𝐴 ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) ∧ (𝑥𝐴𝜓)))
76abbii 2233 . . . 4 {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐴𝜓))}
85, 7eqtr4i 2141 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
94, 8eqtr4i 2141 . 2 {𝑥𝐴 ∣ (𝜑𝜓)} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐴𝜓)})
103, 9eqtr4i 2141 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wcel 1465  {cab 2103  {crab 2397  cin 3040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-in 3047
This theorem is referenced by:  rabnc  3365  iooinsup  11014  phiprmpw  11825  unennn  11837
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