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Theorem ineqan12d 3218
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 3211 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
41, 2, 3syl2an 284 1  |-  ( (
ph  /\  ps )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1296    i^i cin 3012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019
This theorem is referenced by:  fvun1  5405  fndmin  5445  offval  5901  ofrfval  5902  offval3  5943  iooinsup  10820
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