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Theorem intprg 4076
 Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4075. 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 3875 . . . 4
21inteqd 4047 . . 3
3 ineq1 3527 . . 3
42, 3eqeq12d 2449 . 2
5 preq2 3876 . . . 4
65inteqd 4047 . . 3
7 ineq2 3528 . . 3
86, 7eqeq12d 2449 . 2
9 vex 2951 . . 3
10 vex 2951 . . 3
119, 10intpr 4075 . 2
124, 8, 11vtocl2g 3007 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725   cin 3311  cpr 3807  cint 4042 This theorem is referenced by:  intsng  4077  inelfi  7415  mreincl  13816  subrgin  15883  lssincl  16033  incld  17099  difelsiga  24508  inidl  26631 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-un 3317  df-in 3319  df-sn 3812  df-pr 3813  df-int 4043
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