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Theorem intprg 4323
 Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4322. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
Assertion
Ref Expression
intprg

Proof of Theorem intprg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 4111 . . . 4
21inteqd 4293 . . 3
3 ineq1 3689 . . 3
42, 3eqeq12d 2479 . 2
5 preq2 4112 . . . 4
65inteqd 4293 . . 3
7 ineq2 3690 . . 3
86, 7eqeq12d 2479 . 2
9 vex 3112 . . 3
10 vex 3112 . . 3
119, 10intpr 4322 . 2
124, 8, 11vtocl2g 3171 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819   cin 3470  cpr 4034  cint 4288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-un 3476  df-in 3478  df-sn 4033  df-pr 4035  df-int 4289 This theorem is referenced by:  intsng  4324  inelfi  7896  mreincl  15016  subrgin  17579  lssincl  17738  incld  19671  difelsiga  28306  inidl  30632
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