Theorem inteq 3888

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 2702	. . 3 |- ( A = B -> ( A. y e. A x e. y <-> A. y e. B x e. y ) )
2	1	abbidv 2323	. 2 |- ( A = B -> { x | A. y e. A x e. y } = { x | A. y e. B x e. y } )
3		dfint2 3887	. 2 |- |^| A = { x | A. y e. A x e. y }
4		dfint2 3887	. 2 |- |^| B = { x | A. y e. B x e. y }
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1373   {cab 2191   A.wral 2484   |^|cint 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-int 3886

This theorem is referenced by:  inteqi  3889  inteqd  3890  uniintsnr  3921  rint0  3924  intexr  4194  onintexmid  4621  elreldm  4904  elxp5  5171  1stval2  6241  fundmen  6898  xpsnen  6916  fiintim  7028  elfir  7075  fiinopn  14476  bj-intexr  15844