Theorem inteqd 3927

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 3925	. 2 |- ( A = B -> |^| A = |^| B )
3	1, 2	syl 14	1 |- ( ph -> |^| A = |^| B )

Syntax hints:   -> wi 4   = wceq 1395   |^|cint 3922

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 3923

This theorem is referenced by:  intprg  3955  op1stbg  4569  onsucmin  4598  elreldm  4949  elxp5  5216  fniinfv  5691  1stval2  6299  2ndval2  6300  fundmen  6957  xpsnen  6976  fiintim  7089  elfi2  7135  fi0  7138  cardcl  7349  isnumi  7350  cardval3ex  7353  carden2bex  7358  lspfval  14346  lspval  14348  lsppropd  14390  clsfval  14769  clsval  14779