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Theorem difab 3523
 Description: Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difab ({x φ} {x ψ}) = {x (φ ¬ ψ)}

Proof of Theorem difab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-clab 2340 . . 3 (y {x (φ ¬ ψ)} ↔ [y / x](φ ¬ ψ))
2 sban 2069 . . 3 ([y / x](φ ¬ ψ) ↔ ([y / x]φ [y / x] ¬ ψ))
3 df-clab 2340 . . . . 5 (y {x φ} ↔ [y / x]φ)
43bicomi 193 . . . 4 ([y / x]φy {x φ})
5 sbn 2062 . . . . 5 ([y / x] ¬ ψ ↔ ¬ [y / x]ψ)
6 df-clab 2340 . . . . 5 (y {x ψ} ↔ [y / x]ψ)
75, 6xchbinxr 302 . . . 4 ([y / x] ¬ ψ ↔ ¬ y {x ψ})
84, 7anbi12i 678 . . 3 (([y / x]φ [y / x] ¬ ψ) ↔ (y {x φ} ¬ y {x ψ}))
91, 2, 83bitrri 263 . 2 ((y {x φ} ¬ y {x ψ}) ↔ y {x (φ ¬ ψ)})
109difeqri 3387 1 ({x φ} {x ψ}) = {x (φ ¬ ψ)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339   ∖ cdif 3206 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215 This theorem is referenced by:  notab  3525  difrab  3529  notrab  3532
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