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Theorem inab 4197
 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 2061 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
2 df-clab 2778 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑𝜓)} ↔ [𝑦 / 𝑥](𝜑𝜓))
3 df-clab 2778 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 df-clab 2778 . . . 4 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
53, 4anbi12i 626 . . 3 ((𝑦 ∈ {𝑥𝜑} ∧ 𝑦 ∈ {𝑥𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
61, 2, 53bitr4ri 305 . 2 ((𝑦 ∈ {𝑥𝜑} ∧ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑𝜓)})
76ineqri 4106 1 ({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 396   = wceq 1525  [wsb 2044   ∈ wcel 2083  {cab 2777   ∩ cin 3864 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-v 3442  df-in 3872 This theorem is referenced by:  inrab  4201  inrab2  4202  dfrab3  4204  orduniss2  7411  ssenen  8545  hashf1lem2  13666  ballotlem2  31359  fmla0disjsuc  32255  dfiota3  32995  bj-inrab  33822  ptrest  34443  diophin  38875
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