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Theorem difab 3200
Description: Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difab ({xφ} ∖ {xψ}) = {x ∣ (φ ¬ ψ)}

Proof of Theorem difab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-clab 2024 . . 3 (y {x ∣ (φ ¬ ψ)} ↔ [y / x](φ ¬ ψ))
2 sban 1826 . . 3 ([y / x](φ ¬ ψ) ↔ ([y / x]φ [y / x] ¬ ψ))
3 df-clab 2024 . . . . 5 (y {xφ} ↔ [y / x]φ)
43bicomi 123 . . . 4 ([y / x]φy {xφ})
5 sbn 1823 . . . . 5 ([y / x] ¬ ψ ↔ ¬ [y / x]ψ)
6 df-clab 2024 . . . . 5 (y {xψ} ↔ [y / x]ψ)
75, 6xchbinxr 607 . . . 4 ([y / x] ¬ ψ ↔ ¬ y {xψ})
84, 7anbi12i 433 . . 3 (([y / x]φ [y / x] ¬ ψ) ↔ (y {xφ} ¬ y {xψ}))
91, 2, 83bitrri 196 . 2 ((y {xφ} ¬ y {xψ}) ↔ y {x ∣ (φ ¬ ψ)})
109difeqri 3058 1 ({xφ} ∖ {xψ}) = {x ∣ (φ ¬ ψ)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1242   wcel 1390  [wsb 1642  {cab 2023  cdif 2908
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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914
This theorem is referenced by:  notab  3201  difrab  3205  notrab  3208  imadiflem  4921  imadif  4922
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