Theorem ineq2d 3378

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 3372	. 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 3169

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176

This theorem is referenced by:  disjpr2  3702  rint0  3933  riin0  4008  disji2  4046  xpriindim  4829  riinint  4953  reseq2  4968  csbresg  4976  resindm  5015  isoselem  5907  zfz1isolem1  11017  fsumm1  11812  bitsinv1  12358  ennnfonelemhf1o  12869  nninfdclemcl  12904  nninfdclemp1  12906  nninfdc  12909  ressvalsets  12981  ressbasd  12984  ressinbasd  12991  ressressg  12992  restval  13162  mgpress  13778  subrngpropd  14063  subrgpropd  14100  crng2idl  14378  basis1  14604  baspartn  14607  eltg  14609  tgdom  14629  ntrval  14667  resttopon2  14735  restopnb  14738  qtopbasss  15078