Theorem ineq2d 3337

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 3331	. 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 3129

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 2740  df-in 3136

This theorem is referenced by:  disjpr2  3657  rint0  3884  riin0  3959  disji2  3997  xpriindim  4766  riinint  4889  reseq2  4903  csbresg  4911  resindm  4950  isoselem  5821  zfz1isolem1  10820  fsumm1  11424  ennnfonelemhf1o  12414  nninfdclemcl  12449  nninfdclemp1  12451  nninfdc  12454  ressvalsets  12524  ressbasd  12527  ressinbasd  12533  ressressg  12534  restval  12694  mgpress  13141  subrgpropd  13369  basis1  13550  baspartn  13553  eltg  13555  tgdom  13575  ntrval  13613  resttopon2  13681  restopnb  13684  qtopbasss  14024