Theorem ineq2d 3364

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 3358	. 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 1364   i^i cin 3156

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163

This theorem is referenced by:  disjpr2  3686  rint0  3913  riin0  3988  disji2  4026  xpriindim  4804  riinint  4927  reseq2  4941  csbresg  4949  resindm  4988  isoselem  5867  zfz1isolem1  10932  fsumm1  11581  ennnfonelemhf1o  12630  nninfdclemcl  12665  nninfdclemp1  12667  nninfdc  12670  ressvalsets  12742  ressbasd  12745  ressinbasd  12752  ressressg  12753  restval  12916  mgpress  13487  subrngpropd  13772  subrgpropd  13809  crng2idl  14087  basis1  14283  baspartn  14286  eltg  14288  tgdom  14308  ntrval  14346  resttopon2  14414  restopnb  14417  qtopbasss  14757