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Theorem inab 4300
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 2084 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
2 df-clab 2711 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑𝜓)} ↔ [𝑦 / 𝑥](𝜑𝜓))
3 df-clab 2711 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 df-clab 2711 . . . 4 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
53, 4anbi12i 628 . . 3 ((𝑦 ∈ {𝑥𝜑} ∧ 𝑦 ∈ {𝑥𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
61, 2, 53bitr4ri 304 . 2 ((𝑦 ∈ {𝑥𝜑} ∧ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑𝜓)})
76ineqri 4205 1 ({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  [wsb 2068  wcel 2107  {cab 2710  cin 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956
This theorem is referenced by:  inrab  4307  inrab2  4308  dfrab3  4310  orduniss2  7821  ssenen  9151  hashf1lem2  14417  symgsubmefmnd  19266  ballotlem2  33487  fmla0disjsuc  34389  dfiota3  34895  bj-inrab  35807  ptrest  36487  sticksstones22  40984  diophin  41510
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