Theorem inteq 3810

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 2652	. . 3 |- ( A = B -> ( A. y e. A x e. y <-> A. y e. B x e. y ) )
2	1	abbidv 2275	. 2 |- ( A = B -> { x | A. y e. A x e. y } = { x | A. y e. B x e. y } )
3		dfint2 3809	. 2 |- |^| A = { x | A. y e. A x e. y }
4		dfint2 3809	. 2 |- |^| B = { x | A. y e. B x e. y }
5	2, 3, 4	3eqtr4g 2215	1 |- ( A = B -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1335   {cab 2143   A.wral 2435   |^|cint 3807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-int 3808

This theorem is referenced by:  inteqi  3811  inteqd  3812  uniintsnr  3843  rint0  3846  intexr  4111  onintexmid  4531  elreldm  4811  elxp5  5073  1stval2  6100  fundmen  6748  xpsnen  6763  fiintim  6870  elfir  6914  fiinopn  12389  bj-intexr  13470