Theorem ineq2d 3382

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 3376	. 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 1373   i^i cin 3173

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 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180

This theorem is referenced by:  disjpr2  3707  rint0  3938  riin0  4013  disji2  4051  xpriindim  4834  riinint  4958  reseq2  4973  csbresg  4981  resindm  5020  isoselem  5912  zfz1isolem1  11022  fsumm1  11842  bitsinv1  12388  ennnfonelemhf1o  12899  nninfdclemcl  12934  nninfdclemp1  12936  nninfdc  12939  ressvalsets  13011  ressbasd  13014  ressinbasd  13021  ressressg  13022  restval  13192  mgpress  13808  subrngpropd  14093  subrgpropd  14130  crng2idl  14408  basis1  14634  baspartn  14637  eltg  14639  tgdom  14659  ntrval  14697  resttopon2  14765  restopnb  14768  qtopbasss  15108