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Theorem inab 3199
 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 ∣ (φ ψ)}

Proof of Theorem inab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sban 1826 . . 3 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
2 df-clab 2024 . . 3 (y {x ∣ (φ ψ)} ↔ [y / x](φ ψ))
3 df-clab 2024 . . . 4 (y {xφ} ↔ [y / x]φ)
4 df-clab 2024 . . . 4 (y {xψ} ↔ [y / x]ψ)
53, 4anbi12i 433 . . 3 ((y {xφ} y {xψ}) ↔ ([y / x]φ [y / x]ψ))
61, 2, 53bitr4ri 202 . 2 ((y {xφ} y {xψ}) ↔ y {x ∣ (φ ψ)})
76ineqri 3124 1 ({xφ} ∩ {xψ}) = {x ∣ (φ ψ)}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242   ∈ wcel 1390  [wsb 1642  {cab 2023   ∩ cin 2910 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918 This theorem is referenced by:  inrab  3203  inrab2  3204  dfrab2  3206  dfrab3  3207  imainlem  4923  imain  4924
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