Theorem ineq2d 3405

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 3399	. 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 1395   i^i cin 3196

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-v 2801  df-in 3203

This theorem is referenced by:  disjpr2  3730  rint0  3962  riin0  4037  disji2  4075  xpriindim  4860  riinint  4985  reseq2  5000  csbresg  5008  resindm  5047  isoselem  5950  zfz1isolem1  11075  fsumm1  11943  bitsinv1  12489  ennnfonelemhf1o  13000  nninfdclemcl  13035  nninfdclemp1  13037  nninfdc  13040  ressvalsets  13113  ressbasd  13116  ressinbasd  13123  ressressg  13124  restval  13294  mgpress  13910  subrngpropd  14196  subrgpropd  14233  crng2idl  14511  basis1  14737  baspartn  14740  eltg  14742  tgdom  14762  ntrval  14800  resttopon2  14868  restopnb  14871  qtopbasss  15211