Theorem ineq12 3688

Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)

Assertion

Ref	Expression
ineq12	|- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )

Proof of Theorem ineq12

Step	Hyp	Ref	Expression
1		ineq1 3686	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		ineq2 3687	. 2 |- ( C = D -> ( B i^i C ) = ( B i^i D ) )
3	1, 2	sylan9eq 2521	1 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   i^i cin 3468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-in 3476

This theorem is referenced by:  ineq12i  3691  ineq12d  3694  ineqan12d  3695  fnun  5678  undifixp  7495  endisj  7594  sbthlem8  7624  fiin  7871  pm54.43  8370  kmlem9  8527  indistopon  19261  epttop  19269  restbas  19418  ordtbas2  19451  txbas  19796  ptbasin  19806  trfbas2  20072  snfil  20093  fbasrn  20113  trfil2  20116  fmfnfmlem3  20185  ustuqtop2  20473  minveclem3b  21571  isperp  23790  frrlem4  28817  diophin  30161  kelac2lem  30467