Theorem ineq2d 3432

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 3426	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   i^i cin 3219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is used by:  disjpr2  3773  rint0  4009  riin0  4084  disji2  4122  xpriindim  4918  riinint  5043  reseq2  5058  csbresg  5066  resindm  5105  isoselem  6026  zfz1isolem1  11308  fsumm1  12202  bitsinv1  12748  ballotfilemfval  13281  ennnfonelemhf1o  13356  nninfdclemcl  13391  nninfdclemp1  13393  nninfdc  13396  ressvalsets  13470  ressbasd  13474  ressinbasd  13481  ressressg  13482  restval  13652  mgpress  14314  subrngpropd  14608  subrgpropd  14645  crng2idl  14952  basis1  15239  baspartn  15242  eltg  15244  tgdom  15264  ntrval  15302  resttopon2  15370  restopnb  15373  qtopbasss  15713  p1evtxdeqfilem  16718