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Theorem inab 3522
 Description: Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
inab ({x φ} ∩ {x ψ}) = {x (φ ψ)}

Proof of Theorem inab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sban 2069 . . 3 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
2 df-clab 2340 . . 3 (y {x (φ ψ)} ↔ [y / x](φ ψ))
3 df-clab 2340 . . . 4 (y {x φ} ↔ [y / x]φ)
4 df-clab 2340 . . . 4 (y {x ψ} ↔ [y / x]ψ)
53, 4anbi12i 678 . . 3 ((y {x φ} y {x ψ}) ↔ ([y / x]φ [y / x]ψ))
61, 2, 53bitr4ri 269 . 2 ((y {x φ} y {x ψ}) ↔ y {x (φ ψ)})
76ineqri 3449 1 ({x φ} ∩ {x ψ}) = {x (φ ψ)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213 This theorem is referenced by:  inrab  3527  inrab2  3528  dfrab2  3530  dfrab3  3531  evenodddisj  4516  spfinex  4537  pmex  6005  nmembers1lem1  6268
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