MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfrab3 Structured version   Visualization version   GIF version

Theorem dfrab3 4277
Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfrab3 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfrab3
StepHypRef Expression
1 df-rab 3147 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 inab 4270 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴𝜑)}
3 abid2 2957 . . 3 {𝑥𝑥𝐴} = 𝐴
43ineq1i 4184 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑})
51, 2, 43eqtr2i 2850 1 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1533  wcel 2110  {cab 2799  {crab 3142  cin 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-in 3942
This theorem is referenced by:  dfrab2  4278  notrab  4279  dfrab3ss  4280  dfif3  4480  dffr3  5956  dfse2  5957  tz6.26  6173  rabfi  8737  dfsup2  8902  ressmplbas2  20230  clsocv  23847  hasheuni  31339  bj-inrab3  34242  bj-reabeq  34334  hashnzfz  40645
  Copyright terms: Public domain W3C validator