Theorem ineq2d 3336

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 3330	. 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 1353   i^i cin 3128

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135

This theorem is referenced by:  disjpr2  3655  rint0  3881  riin0  3955  disji2  3993  xpriindim  4761  riinint  4884  reseq2  4898  csbresg  4906  resindm  4945  isoselem  5815  zfz1isolem1  10804  fsumm1  11408  ennnfonelemhf1o  12397  nninfdclemcl  12432  nninfdclemp1  12434  nninfdc  12437  ressvalsets  12506  ressbasd  12509  ressinbasd  12515  ressressg  12516  restval  12642  basis1  13212  baspartn  13215  eltg  13219  tgdom  13239  ntrval  13277  resttopon2  13345  restopnb  13348  qtopbasss  13688