ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ineq1d Unicode version

Theorem ineq1d 3137
Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
ineq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ineq1d  |-  ( ph  ->  ( A  i^i  C
)  =  ( B  i^i  C ) )

Proof of Theorem ineq1d
StepHypRef Expression
1 ineq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ineq1 3131 . 2  |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C
) )
31, 2syl 14 1  |-  ( ph  ->  ( A  i^i  C
)  =  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    i^i cin 2916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924
This theorem is referenced by:  diftpsn3  3505  ordpwsucexmid  4294  riinint  4593  fnresdisj  5009  fnimadisj  5019  ecinxp  6181  fzval2  8875  fvinim0ffz  9094
  Copyright terms: Public domain W3C validator