Theorem ineq2d 3426

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 3420	. 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 1398   i^i cin 3213

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220

This theorem is referenced by:  disjpr2  3758  rint0  3993  riin0  4068  disji2  4106  xpriindim  4898  riinint  5023  reseq2  5038  csbresg  5046  resindm  5085  isoselem  5999  zfz1isolem1  11240  fsumm1  12131  bitsinv1  12677  ballotfilemfval  13177  ennnfonelemhf1o  13252  nninfdclemcl  13287  nninfdclemp1  13289  nninfdc  13292  ressvalsets  13365  ressbasd  13368  ressinbasd  13375  ressressg  13376  restval  13546  mgpress  14174  subrngpropd  14466  subrgpropd  14503  crng2idl  14809  basis1  15042  baspartn  15045  eltg  15047  tgdom  15067  ntrval  15105  resttopon2  15173  restopnb  15176  qtopbasss  15516  p1evtxdeqfilem  16436