Theorem ineq2d 3410

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 3404	. 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 1398   i^i cin 3200

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207

This theorem is referenced by:  disjpr2  3737  rint0  3972  riin0  4047  disji2  4085  xpriindim  4874  riinint  4999  reseq2  5014  csbresg  5022  resindm  5061  isoselem  5971  zfz1isolem1  11167  fsumm1  12057  bitsinv1  12603  ennnfonelemhf1o  13114  nninfdclemcl  13149  nninfdclemp1  13151  nninfdc  13154  ressvalsets  13227  ressbasd  13230  ressinbasd  13237  ressressg  13238  restval  13408  mgpress  14025  subrngpropd  14311  subrgpropd  14348  crng2idl  14627  basis1  14858  baspartn  14861  eltg  14863  tgdom  14883  ntrval  14921  resttopon2  14989  restopnb  14992  qtopbasss  15332  p1evtxdeqfilem  16252