Theorem ineq2d 3374

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 3368	. 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 3165

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  disjpr2  3697  rint0  3924  riin0  3999  disji2  4037  xpriindim  4817  riinint  4940  reseq2  4955  csbresg  4963  resindm  5002  isoselem  5891  zfz1isolem1  10987  fsumm1  11760  bitsinv1  12306  ennnfonelemhf1o  12817  nninfdclemcl  12852  nninfdclemp1  12854  nninfdc  12857  ressvalsets  12929  ressbasd  12932  ressinbasd  12939  ressressg  12940  restval  13110  mgpress  13726  subrngpropd  14011  subrgpropd  14048  crng2idl  14326  basis1  14552  baspartn  14555  eltg  14557  tgdom  14577  ntrval  14615  resttopon2  14683  restopnb  14686  qtopbasss  15026