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Theorem inab 4096
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 2558 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
2 df-clab 2793 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑𝜓)} ↔ [𝑦 / 𝑥](𝜑𝜓))
3 df-clab 2793 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 df-clab 2793 . . . 4 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
53, 4anbi12i 614 . . 3 ((𝑦 ∈ {𝑥𝜑} ∧ 𝑦 ∈ {𝑥𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
61, 2, 53bitr4ri 295 . 2 ((𝑦 ∈ {𝑥𝜑} ∧ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑𝜓)})
76ineqri 4005 1 ({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  [wsb 2060  wcel 2156  {cab 2792  cin 3768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-in 3776
This theorem is referenced by:  inrab  4100  inrab2  4101  dfrab3  4103  orduniss2  7259  ssenen  8369  hashf1lem2  13453  ballotlem2  30871  dfiota3  32346  bj-inrab  33229  ptrest  33716  diophin  37832
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