ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difeq2d Unicode version

Theorem difeq2d 3164
Description: Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
Hypothesis
Ref Expression
difeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
difeq2d  |-  ( ph  ->  ( C  \  A
)  =  ( C 
\  B ) )

Proof of Theorem difeq2d
StepHypRef Expression
1 difeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 difeq2 3158 . 2  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
31, 2syl 14 1  |-  ( ph  ->  ( C  \  A
)  =  ( C 
\  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    \ cdif 3038
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-in1 588  ax-in2 589  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-rab 2402  df-dif 3043
This theorem is referenced by:  difeq12d  3165  phplem3  6716  phplem4  6717  phplem3g  6718  phplem4dom  6724  phplem4on  6729  fidifsnen  6732  xpfi  6786  sbthlem2  6814  sbthlemi3  6815  isbth  6823  ismkvnex  6997  setsvalg  11916  setsvala  11917  exmid1stab  13122
  Copyright terms: Public domain W3C validator