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Theorem notrab 3532
 Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
notrab (A {x A φ}) = {x A ¬ φ}
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem notrab
StepHypRef Expression
1 difab 3523 . 2 ({x x A} {x φ}) = {x (x A ¬ φ)}
2 difin 3492 . . 3 (A (A ∩ {x φ})) = (A {x φ})
3 dfrab3 3531 . . . 4 {x A φ} = (A ∩ {x φ})
43difeq2i 3382 . . 3 (A {x A φ}) = (A (A ∩ {x φ}))
5 abid2 2470 . . . 4 {x x A} = A
65difeq1i 3381 . . 3 ({x x A} {x φ}) = (A {x φ})
72, 4, 63eqtr4i 2383 . 2 (A {x A φ}) = ({x x A} {x φ})
8 df-rab 2623 . 2 {x A ¬ φ} = {x (x A ¬ φ)}
91, 7, 83eqtr4i 2383 1 (A {x A φ}) = {x A ¬ φ}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618   ∖ cdif 3206   ∩ 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  df-dif 3215 This theorem is referenced by: (None)
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