Theorem inteq 3774

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 2626	. . 3 |- ( A = B -> ( A. y e. A x e. y <-> A. y e. B x e. y ) )
2	1	abbidv 2257	. 2 |- ( A = B -> { x | A. y e. A x e. y } = { x | A. y e. B x e. y } )
3		dfint2 3773	. 2 |- |^| A = { x | A. y e. A x e. y }
4		dfint2 3773	. 2 |- |^| B = { x | A. y e. B x e. y }
5	2, 3, 4	3eqtr4g 2197	1 |- ( A = B -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1331   {cab 2125   A.wral 2416   |^|cint 3771

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-int 3772

This theorem is referenced by:  inteqi  3775  inteqd  3776  uniintsnr  3807  rint0  3810  intexr  4075  onintexmid  4487  elreldm  4765  elxp5  5027  1stval2  6053  fundmen  6700  xpsnen  6715  fiintim  6817  elfir  6861  fiinopn  12171  bj-intexr  13106