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Theorem ineq12d 3273
Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
ineq1d.1  |-  ( ph  ->  A  =  B )
ineq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
ineq12d  |-  ( ph  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )

Proof of Theorem ineq12d
StepHypRef Expression
1 ineq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ineq12d.2 . 2  |-  ( ph  ->  C  =  D )
3 ineq12 3267 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
41, 2, 3syl2anc 408 1  |-  ( ph  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    i^i cin 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072
This theorem is referenced by:  csbing  3278  funprg  5168  funtpg  5169  offval  5982  ofrfval  5983  undifdc  6805  djudom  6971  ressid2  12007  ressval2  12008
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