ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfrab2 GIF version

Theorem dfrab2 3212
Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
dfrab2 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfrab2
StepHypRef Expression
1 df-rab 2315 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 inab 3205 . . 3 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴𝜑)}
3 abid2 2158 . . . 4 {𝑥𝑥𝐴} = 𝐴
43ineq1i 3134 . . 3 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑})
52, 4eqtr3i 2062 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = (𝐴 ∩ {𝑥𝜑})
6 incom 3129 . 2 (𝐴 ∩ {𝑥𝜑}) = ({𝑥𝜑} ∩ 𝐴)
71, 5, 63eqtri 2064 1 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
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: (None)
  Copyright terms: Public domain W3C validator