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Theorem dfrab2 3530
 Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
dfrab2 {x A φ} = ({x φ} ∩ A)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem dfrab2
StepHypRef Expression
1 df-rab 2623 . 2 {x A φ} = {x (x A φ)}
2 inab 3522 . . 3 ({x x A} ∩ {x φ}) = {x (x A φ)}
3 abid2 2470 . . . 4 {x x A} = A
43ineq1i 3453 . . 3 ({x x A} ∩ {x φ}) = (A ∩ {x φ})
52, 4eqtr3i 2375 . 2 {x (x A φ)} = (A ∩ {x φ})
6 incom 3448 . 2 (A ∩ {x φ}) = ({x φ} ∩ A)
71, 5, 63eqtri 2377 1 {x A φ} = ({x φ} ∩ A)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213 This theorem is referenced by:  phialllem1  4616  frds  5935  nenpw1pwlem1  6084  nmembers1lem1  6268
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