Theorem inteq 3930

Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999.)

Assertion

Ref	Expression
inteq	|- (A = B -> |^|A = |^|B)

Proof of Theorem inteq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 2808	. . 3 |- (A = B -> (A.y e. A x e. y <-> A.y e. B x e. y))
2	1	abbidv 2468	. 2 |- (A = B -> {x | A.y e. A x e. y} = {x | A.y e. B x e. y})
3		dfint2 3929	. 2 |- |^|A = {x | A.y e. A x e. y}
4		dfint2 3929	. 2 |- |^|B = {x | A.y e. B x e. y}
5	2, 3, 4	3eqtr4g 2410	1 |- (A = B -> |^|A = |^|B)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  |^|cint 3927

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-int 3928

This theorem is referenced by:  inteqi  3931  inteqd  3932  unissint  3951  uniintsn  3964  rint0  3967  clos1eq1  5875  clos1eq2  5876