Theorem inteq 3973

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 2749	. . 3 |- ( A = B -> ( A. y e. A x e. y <-> A. y e. B x e. y ) )
2	1	abbidv 2358	. 2 |- ( A = B -> { x | A. y e. A x e. y } = { x | A. y e. B x e. y } )
3		dfint2 3972	. 2 |- |^| A = { x | A. y e. A x e. y }
4		dfint2 3972	. 2 |- |^| B = { x | A. y e. B x e. y }
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> |^| A = |^| B )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   {cab 2224   A.wral 2528   |^|cint 3970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-int 3971

This theorem is used by:  inteqi  3974  inteqd  3975  uniintsnr  4006  rint0  4009  intexr  4286  onintexmid  4720  elreldm  5008  elxp5  5276  1stval2  6389  fundmen  7094  xpsnen  7119  fiintim  7238  elfir  7307  fiinopn  15105  bj-intexr  16934