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Theorem inrab2 3210
 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 2315 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 abid2 2158 . . . 4 {𝑥𝑥𝐵} = 𝐵
32eqcomi 2044 . . 3 𝐵 = {𝑥𝑥𝐵}
41, 3ineq12i 3136 . 2 ({𝑥𝐴𝜑} ∩ 𝐵) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵})
5 df-rab 2315 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)}
6 inab 3205 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ 𝑥𝐵)}
7 elin 3126 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
87anbi1i 431 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
9 an32 496 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ 𝑥𝐵))
108, 9bitri 173 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ 𝑥𝐵))
1110abbii 2153 . . . 4 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ 𝑥𝐵)}
126, 11eqtr4i 2063 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵}) = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)}
135, 12eqtr4i 2063 . 2 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥𝑥𝐵})
144, 13eqtr4i 2063 1 ({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {cab 2026  {crab 2310   ∩ cin 2916 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-in 2924 This theorem is referenced by:  iooval2  8784  fzval2  8877
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