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Theorem difab 4301
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 2706 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)} ↔ [𝑦 / 𝑥](𝜑 ∧ ¬ 𝜓))
2 sban 2076 . . 3 ([𝑦 / 𝑥](𝜑 ∧ ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥] ¬ 𝜓))
3 df-clab 2706 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
43bicomi 223 . . . 4 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
5 sbn 2270 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑦 / 𝑥]𝜓)
6 df-clab 2706 . . . . 5 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
75, 6xchbinxr 335 . . . 4 ([𝑦 / 𝑥] ¬ 𝜓 ↔ ¬ 𝑦 ∈ {𝑥𝜓})
84, 7anbi12i 627 . . 3 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥] ¬ 𝜓) ↔ (𝑦 ∈ {𝑥𝜑} ∧ ¬ 𝑦 ∈ {𝑥𝜓}))
91, 2, 83bitrri 298 . 2 ((𝑦 ∈ {𝑥𝜑} ∧ ¬ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)})
109difeqri 4122 1 ({𝑥𝜑} ∖ {𝑥𝜓}) = {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1534  [wsb 2060  wcel 2099  {cab 2705  cdif 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-dif 3950
This theorem is referenced by:  notab  4305  difrab  4309  notrab  4312
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