Theorem inteqd 3896

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 3894	. 2 |- ( A = B -> |^| A = |^| B )
3	1, 2	syl 14	1 |- ( ph -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1373   |^|cint 3891

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-int 3892

This theorem is referenced by:  intprg  3924  op1stbg  4534  onsucmin  4563  elreldm  4913  elxp5  5180  fniinfv  5650  1stval2  6254  2ndval2  6255  fundmen  6912  xpsnen  6931  fiintim  7043  elfi2  7089  fi0  7092  cardcl  7303  isnumi  7304  cardval3ex  7307  carden2bex  7312  lspfval  14225  lspval  14227  lsppropd  14269  clsfval  14648  clsval  14658