Theorem inteqd 3876

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 3874	. 2 |- ( A = B -> |^| A = |^| B )
3	1, 2	syl 14	1 |- ( ph -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1364   |^|cint 3871

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 3872

This theorem is referenced by:  intprg  3904  op1stbg  4511  onsucmin  4540  elreldm  4889  elxp5  5155  fniinfv  5616  1stval2  6210  2ndval2  6211  fundmen  6862  xpsnen  6877  fiintim  6987  elfi2  7033  fi0  7036  cardcl  7243  isnumi  7244  cardval3ex  7247  carden2bex  7251  lspfval  13887  lspval  13889  lsppropd  13931  clsfval  14280  clsval  14290