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Theorem inrab2 3528
 Description: Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
Assertion
Ref Expression
inrab2 ({x A φ} ∩ B) = {x (AB) φ}
Distinct variable group:   x,B
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem inrab2
StepHypRef Expression
1 df-rab 2623 . . 3 {x A φ} = {x (x A φ)}
2 abid2 2470 . . . 4 {x x B} = B
32eqcomi 2357 . . 3 B = {x x B}
41, 3ineq12i 3455 . 2 ({x A φ} ∩ B) = ({x (x A φ)} ∩ {x x B})
5 df-rab 2623 . . 3 {x (AB) φ} = {x (x (AB) φ)}
6 inab 3522 . . . 4 ({x (x A φ)} ∩ {x x B}) = {x ((x A φ) x B)}
7 elin 3219 . . . . . . 7 (x (AB) ↔ (x A x B))
87anbi1i 676 . . . . . 6 ((x (AB) φ) ↔ ((x A x B) φ))
9 an32 773 . . . . . 6 (((x A x B) φ) ↔ ((x A φ) x B))
108, 9bitri 240 . . . . 5 ((x (AB) φ) ↔ ((x A φ) x B))
1110abbii 2465 . . . 4 {x (x (AB) φ)} = {x ((x A φ) x B)}
126, 11eqtr4i 2376 . . 3 ({x (x A φ)} ∩ {x x B}) = {x (x (AB) φ)}
135, 12eqtr4i 2376 . 2 {x (AB) φ} = ({x (x A φ)} ∩ {x x B})
144, 13eqtr4i 2376 1 ({x A φ} ∩ B) = {x (AB) φ}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618   ∩ 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-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213 This theorem is referenced by: (None)
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