MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ineqan12d Unicode version

Theorem ineqan12d 3487
Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
Hypotheses
Ref Expression
ineq1d.1  |-  ( ph  ->  A  =  B )
ineqan12d.2  |-  ( ps 
->  C  =  D
)
Assertion
Ref Expression
ineqan12d  |-  ( (
ph  /\  ps )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )

Proof of Theorem ineqan12d
StepHypRef Expression
1 ineq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ineqan12d.2 . 2  |-  ( ps 
->  C  =  D
)
3 ineq12 3480 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
41, 2, 3syl2an 464 1  |-  ( (
ph  /\  ps )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    i^i cin 3262
This theorem is referenced by:  fvun1  5733  fndmin  5776  offval  6251  ofrfval  6252  offval3  6257  fpar  6389  fisn  7367  ixxin  10865  vdwmc  13273  cssincl  16838  inmbl  19303  iundisj2  19310  itg1addlem3  19457  fh1  22968  iundisj2f  23873  iundisj2fi  23991  wfrlem4  25283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-in 3270
  Copyright terms: Public domain W3C validator