Theorem inteq 3957

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 2743	. . 3 |- ( A = B -> ( A. y e. A x e. y <-> A. y e. B x e. y ) )
2	1	abbidv 2354	. 2 |- ( A = B -> { x | A. y e. A x e. y } = { x | A. y e. B x e. y } )
3		dfint2 3956	. 2 |- |^| A = { x | A. y e. A x e. y }
4		dfint2 3956	. 2 |- |^| B = { x | A. y e. B x e. y }
5	2, 3, 4	3eqtr4g 2292	1 |- ( A = B -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1398   {cab 2220   A.wral 2522   |^|cint 3954

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-int 3955

This theorem is referenced by:  inteqi  3958  inteqd  3959  uniintsnr  3990  rint0  3993  intexr  4267  onintexmid  4700  elreldm  4988  elxp5  5256  1stval2  6362  fundmen  7060  xpsnen  7085  fiintim  7204  elfir  7273  fiinopn  14995  bj-intexr  16804