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Theorem inteq 3929
 Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
Assertion
Ref Expression
inteq (A = BA = B)

Proof of Theorem inteq
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 2807 . . 3 (A = B → (y A x yy B x y))
21abbidv 2467 . 2 (A = B → {x y A x y} = {x y B x y})
3 dfint2 3928 . 2 A = {x y A x y}
4 dfint2 3928 . 2 B = {x y B x y}
52, 3, 43eqtr4g 2410 1 (A = BA = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-int 3927 This theorem is referenced by:  inteqi  3930  inteqd  3931  unissint  3950  uniintsn  3963  rint0  3966  clos1eq1  5874  clos1eq2  5875
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