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  4920  riinint  5045  reseq2  5060  csbresg  5068  resindm  5108  isoselem  6029  ofovex  6311  zfz1isolem1  11314  fsumm1  12208  bitsinv1  12754  ballotfilemfval  13287  ennnfonelemhf1o  13362  nninfdclemcl  13397  nninfdclemp1  13399  nninfdc  13402  ressvalsets  13476  ressbasd  13480  ressinbasd  13487  ressressg  13488  restval  13658  mgpress  14337  subrngpropd  14633  subrgpropd  14670  crng2idl  14977  basis1  15273  baspartn  15276  eltg  15278  tgdom  15298  ntrval  15336  resttopon2  15404  restopnb  15407  qtopbasss  15747  p1evtxdeqfilem  16757