Theorem inteqd 3928

Description: Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)

Hypothesis

Ref	Expression
inteqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
inteqd	|- ( ph -> |^| A = |^| B )

Proof of Theorem inteqd

Step	Hyp	Ref	Expression
1		inteqd.1	. 2 |- ( ph -> A = B )
2		inteq 3926	. 2 |- ( A = B -> |^| A = |^| B )
3	1, 2	syl 14	1 |- ( ph -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1395   |^|cint 3923

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-int 3924

This theorem is referenced by:  intprg  3956  op1stbg  4570  onsucmin  4599  elreldm  4950  elxp5  5217  fniinfv  5694  1stval2  6307  2ndval2  6308  fundmen  6967  xpsnen  6988  fiintim  7104  elfi2  7150  fi0  7153  cardcl  7364  isnumi  7365  cardval3ex  7368  carden2bex  7373  lspfval  14367  lspval  14369  lsppropd  14411  clsfval  14790  clsval  14800