Theorem ineq2 2211

Description: Equality theorem for intersection of two classes.

Assertion

Ref	Expression
ineq2	|- (A = B -> (C i^i A) = (C i^i B))

Proof of Theorem ineq2

Step	Hyp	Ref	Expression
1		ineq1 2210	. 2 |- (A = B -> (A i^i C) = (B i^i C))
2		incom 2208	. 2 |- (C i^i A) = (A i^i C)
3		incom 2208	. 2 |- (C i^i B) = (B i^i C)
4	1, 2, 3	3eqtr4g 1531	1 |- (A = B -> (C i^i A) = (C i^i B))

Syntax hints:   -> wi 3   = wceq 956   i^i cin 2046

This theorem is referenced by:  ineq12 2212  ineq2i 2214  ineq2d 2217  uneqin 2256  wefrc 2943  onfr 2986  fiint 4559  fiintOLD 4560  cplem2 4721  aceq5 4740  kmlem2 4766  kmlem13 4777  kmlem14 4778  inopnt 7600  basis1t 7614  basis2t 7615  sn0top 7647  fctopOLD 7650  cctop 7652  metres 7823  methausi 7881  shinclt 9351  chinclt 9422  chdmm1t 9448  ledit 9463  cmbrt 9527  cmbr3 9543  cmbr3t 9551  pjoml2t 9554  stcltrlem1 10203  mdbrt 10221  dmdbrt 10226  cvmdt 10263  cvexcht 10301  sumdmdi 10342  mddmdin0 10358  infi1 10447  infi1OLD 10448  intprd 10471  neiopne 10474  subsp 10554  filint 10562  filintf 10569  cnfilca 10583  cnfilcaOLD 10584  dtt2 10618  ishgrag 10769  hgralem 10770  ispgrag 10779

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051

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