Theorem inteq 3887

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 2701	. . 3 |- ( A = B -> ( A. y e. A x e. y <-> A. y e. B x e. y ) )
2	1	abbidv 2322	. 2 |- ( A = B -> { x | A. y e. A x e. y } = { x | A. y e. B x e. y } )
3		dfint2 3886	. 2 |- |^| A = { x | A. y e. A x e. y }
4		dfint2 3886	. 2 |- |^| B = { x | A. y e. B x e. y }
5	2, 3, 4	3eqtr4g 2262	1 |- ( A = B -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1372   {cab 2190   A.wral 2483   |^|cint 3884

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-int 3885

This theorem is referenced by:  inteqi  3888  inteqd  3889  uniintsnr  3920  rint0  3923  intexr  4193  onintexmid  4620  elreldm  4903  elxp5  5170  1stval2  6240  fundmen  6897  xpsnen  6915  fiintim  7027  elfir  7074  fiinopn  14447  bj-intexr  15806