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  3656  rint0  3883  riin0  3958  disji2  3996  xpriindim  4765  riinint  4888  reseq2  4902  csbresg  4910  resindm  4949  isoselem  5820  zfz1isolem1  10819  fsumm1  11423  ennnfonelemhf1o  12413  nninfdclemcl  12448  nninfdclemp1  12450  nninfdc  12453  ressvalsets  12523  ressbasd  12526  ressinbasd  12532  ressressg  12533  restval  12693  mgpress  13139  subrgpropd  13367  basis1  13517  baspartn  13520  eltg  13522  tgdom  13542  ntrval  13580  resttopon2  13648  restopnb  13651  qtopbasss  13991