ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpeq12d Unicode version

Theorem xpeq12d 4463
Description: Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.)
Hypotheses
Ref Expression
xpeq1d.1  |-  ( ph  ->  A  =  B )
xpeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
xpeq12d  |-  ( ph  ->  ( A  X.  C
)  =  ( B  X.  D ) )

Proof of Theorem xpeq12d
StepHypRef Expression
1 xpeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 xpeq12d.2 . 2  |-  ( ph  ->  C  =  D )
3 xpeq12 4457 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
41, 2, 3syl2anc 403 1  |-  ( ph  ->  ( A  X.  C
)  =  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    X. cxp 4436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-opab 3900  df-xp 4444
This theorem is referenced by:  opeliunxp  4493  mpt2mptsx  5967  dmmpt2ssx  5969  fmpt2x  5970  disjxp1  6001  erssxp  6315  fsum2dlemstep  10828  fisumcom2  10832
  Copyright terms: Public domain W3C validator