Theorem inteq 3873

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 2690	. . 3 |- ( A = B -> ( A. y e. A x e. y <-> A. y e. B x e. y ) )
2	1	abbidv 2311	. 2 |- ( A = B -> { x | A. y e. A x e. y } = { x | A. y e. B x e. y } )
3		dfint2 3872	. 2 |- |^| A = { x | A. y e. A x e. y }
4		dfint2 3872	. 2 |- |^| B = { x | A. y e. B x e. y }
5	2, 3, 4	3eqtr4g 2251	1 |- ( A = B -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1364   {cab 2179   A.wral 2472   |^|cint 3870

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-int 3871

This theorem is referenced by:  inteqi  3874  inteqd  3875  uniintsnr  3906  rint0  3909  intexr  4179  onintexmid  4605  elreldm  4888  elxp5  5154  1stval2  6208  fundmen  6860  xpsnen  6875  fiintim  6985  elfir  7032  fiinopn  14172  bj-intexr  15400