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Theorem inrab2 3319
Description: Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
Assertion
Ref Expression
inrab2 ({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem inrab2
StepHypRef Expression
1 df-rab 2402 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 abid2 2238 . . . 4 {𝑥𝑥𝐵} = 𝐵
32eqcomi 2121 . . 3 𝐵 = {𝑥𝑥𝐵}
41, 3ineq12i 3245 . 2 ({𝑥𝐴𝜑} ∩ 𝐵) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵})
5 df-rab 2402 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)}
6 inab 3314 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ 𝑥𝐵)}
7 elin 3229 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
87anbi1i 453 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
9 an32 536 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ 𝑥𝐵))
108, 9bitri 183 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ 𝑥𝐵))
1110abbii 2233 . . . 4 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ 𝑥𝐵)}
126, 11eqtr4i 2141 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵}) = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)}
135, 12eqtr4i 2141 . 2 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵})
144, 13eqtr4i 2141 1 ({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
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:  iooval2  9666  fzval2  9761  dfphi2  11823  znnen  11838
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