Theorem inteq 3849

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 2673	. . 3 |- ( A = B -> ( A. y e. A x e. y <-> A. y e. B x e. y ) )
2	1	abbidv 2295	. 2 |- ( A = B -> { x | A. y e. A x e. y } = { x | A. y e. B x e. y } )
3		dfint2 3848	. 2 |- |^| A = { x | A. y e. A x e. y }
4		dfint2 3848	. 2 |- |^| B = { x | A. y e. B x e. y }
5	2, 3, 4	3eqtr4g 2235	1 |- ( A = B -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1353   {cab 2163   A.wral 2455   |^|cint 3846

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-int 3847

This theorem is referenced by:  inteqi  3850  inteqd  3851  uniintsnr  3882  rint0  3885  intexr  4152  onintexmid  4574  elreldm  4855  elxp5  5119  1stval2  6158  fundmen  6808  xpsnen  6823  fiintim  6930  elfir  6974  fiinopn  13589  bj-intexr  14745