Theorem ineq2d 3360

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 3354	. 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 1364   i^i cin 3152

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159

This theorem is referenced by:  disjpr2  3682  rint0  3909  riin0  3984  disji2  4022  xpriindim  4800  riinint  4923  reseq2  4937  csbresg  4945  resindm  4984  isoselem  5863  zfz1isolem1  10911  fsumm1  11559  ennnfonelemhf1o  12570  nninfdclemcl  12605  nninfdclemp1  12607  nninfdc  12610  ressvalsets  12682  ressbasd  12685  ressinbasd  12692  ressressg  12693  restval  12856  mgpress  13427  subrngpropd  13712  subrgpropd  13749  crng2idl  14027  basis1  14215  baspartn  14218  eltg  14220  tgdom  14240  ntrval  14278  resttopon2  14346  restopnb  14349  qtopbasss  14689