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Theorem inab 3850
Description: Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
inab ({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Proof of Theorem inab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sban 2383 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
2 df-clab 2593 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑𝜓)} ↔ [𝑦 / 𝑥](𝜑𝜓))
3 df-clab 2593 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 df-clab 2593 . . . 4 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
53, 4anbi12i 728 . . 3 ((𝑦 ∈ {𝑥𝜑} ∧ 𝑦 ∈ {𝑥𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
61, 2, 53bitr4ri 291 . 2 ((𝑦 ∈ {𝑥𝜑} ∧ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑𝜓)})
76ineqri 3764 1 ({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  [wsb 1866  wcel 1976  {cab 2592  cin 3535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-v 3171  df-in 3543
This theorem is referenced by:  inrab  3854  inrab2  3855  dfrab3  3857  orduniss2  6899  ssenen  7993  hashf1lem2  13046  ballotlem2  29680  dfiota3  31003  bj-inrab  31915  ptrest  32378  diophin  36154
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