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Theorem difab 3929
 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 2638 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)} ↔ [𝑦 / 𝑥](𝜑 ∧ ¬ 𝜓))
2 sban 2427 . . 3 ([𝑦 / 𝑥](𝜑 ∧ ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥] ¬ 𝜓))
3 df-clab 2638 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
43bicomi 214 . . . 4 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
5 sbn 2419 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑦 / 𝑥]𝜓)
6 df-clab 2638 . . . . 5 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
75, 6xchbinxr 324 . . . 4 ([𝑦 / 𝑥] ¬ 𝜓 ↔ ¬ 𝑦 ∈ {𝑥𝜓})
84, 7anbi12i 733 . . 3 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥] ¬ 𝜓) ↔ (𝑦 ∈ {𝑥𝜑} ∧ ¬ 𝑦 ∈ {𝑥𝜓}))
91, 2, 83bitrri 287 . 2 ((𝑦 ∈ {𝑥𝜑} ∧ ¬ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)})
109difeqri 3763 1 ({𝑥𝜑} ∖ {𝑥𝜓}) = {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1523  [wsb 1937   ∈ wcel 2030  {cab 2637   ∖ cdif 3604 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610 This theorem is referenced by:  notab  3930  difrab  3934  notrab  3937
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