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Theorem ineqan12d 3536
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 3529 . 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 1652    i^i cin 3311
This theorem is referenced by:  fvun1  5786  fndmin  5829  offval  6304  ofrfval  6305  offval3  6310  fpar  6442  fisn  7424  ixxin  10925  vdwmc  13338  cssincl  16907  inmbl  19428  iundisj2  19435  itg1addlem3  19582  fh1  23112  iundisj2f  24022  iundisj2fi  24145  wfrlem4  25533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319
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