Theorem inteqd 3889

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 3887	. 2 |- ( A = B -> |^| A = |^| B )
3	1, 2	syl 14	1 |- ( ph -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1372   |^|cint 3884

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-int 3885

This theorem is referenced by:  intprg  3917  op1stbg  4525  onsucmin  4554  elreldm  4903  elxp5  5170  fniinfv  5636  1stval2  6240  2ndval2  6241  fundmen  6897  xpsnen  6915  fiintim  7027  elfi2  7073  fi0  7076  cardcl  7287  isnumi  7288  cardval3ex  7291  carden2bex  7296  lspfval  14092  lspval  14094  lsppropd  14136  clsfval  14515  clsval  14525