Theorem ineq2d 3322

Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Hypothesis

Ref	Expression
ineq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
ineq2d	|- ( ph -> ( C i^i A ) = ( C i^i B ) )

Proof of Theorem ineq2d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 |- ( ph -> A = B )
2		ineq2 3316	. 2 |- ( A = B -> ( C i^i A ) = ( C i^i B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C i^i A ) = ( C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1343   i^i cin 3114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-in 3121

This theorem is referenced by:  disjpr2  3639  rint0  3862  riin0  3936  disji2  3974  xpriindim  4741  riinint  4864  reseq2  4878  csbresg  4886  resindm  4925  isoselem  5787  zfz1isolem1  10749  fsumm1  11353  ennnfonelemhf1o  12342  nninfdclemcl  12377  nninfdclemp1  12379  nninfdc  12382  restval  12557  basis1  12645  baspartn  12648  eltg  12652  tgdom  12672  ntrval  12710  resttopon2  12778  restopnb  12781  qtopbasss  13121