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Theorem difab 3419
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 2176 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)} ↔ [𝑦 / 𝑥](𝜑 ∧ ¬ 𝜓))
2 sban 1967 . . 3 ([𝑦 / 𝑥](𝜑 ∧ ¬ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥] ¬ 𝜓))
3 df-clab 2176 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
43bicomi 132 . . . 4 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
5 sbn 1964 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑦 / 𝑥]𝜓)
6 df-clab 2176 . . . . 5 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
75, 6xchbinxr 684 . . . 4 ([𝑦 / 𝑥] ¬ 𝜓 ↔ ¬ 𝑦 ∈ {𝑥𝜓})
84, 7anbi12i 460 . . 3 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥] ¬ 𝜓) ↔ (𝑦 ∈ {𝑥𝜑} ∧ ¬ 𝑦 ∈ {𝑥𝜓}))
91, 2, 83bitrri 207 . 2 ((𝑦 ∈ {𝑥𝜑} ∧ ¬ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)})
109difeqri 3270 1 ({𝑥𝜑} ∖ {𝑥𝜓}) = {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104   = wceq 1364  [wsb 1773  wcel 2160  {cab 2175  cdif 3141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146
This theorem is referenced by:  notab  3420  difrab  3424  notrab  3427  imadiflem  5314  imadif  5315
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