Theorem ineq2d 3374

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 3368	. 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 1373   i^i cin 3165

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  disjpr2  3697  rint0  3924  riin0  3999  disji2  4037  xpriindim  4816  riinint  4939  reseq2  4954  csbresg  4962  resindm  5001  isoselem  5889  zfz1isolem1  10985  fsumm1  11727  bitsinv1  12273  ennnfonelemhf1o  12784  nninfdclemcl  12819  nninfdclemp1  12821  nninfdc  12824  ressvalsets  12896  ressbasd  12899  ressinbasd  12906  ressressg  12907  restval  13077  mgpress  13693  subrngpropd  13978  subrgpropd  14015  crng2idl  14293  basis1  14519  baspartn  14522  eltg  14524  tgdom  14544  ntrval  14582  resttopon2  14650  restopnb  14653  qtopbasss  14993