ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sylan9req Unicode version

Theorem sylan9req 2153
Description: An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
Hypotheses
Ref Expression
sylan9req.1  |-  ( ph  ->  B  =  A )
sylan9req.2  |-  ( ps 
->  B  =  C
)
Assertion
Ref Expression
sylan9req  |-  ( (
ph  /\  ps )  ->  A  =  C )

Proof of Theorem sylan9req
StepHypRef Expression
1 sylan9req.1 . . 3  |-  ( ph  ->  B  =  A )
21eqcomd 2105 . 2  |-  ( ph  ->  A  =  B )
3 sylan9req.2 . 2  |-  ( ps 
->  B  =  C
)
42, 3sylan9eq 2152 1  |-  ( (
ph  /\  ps )  ->  A  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1299
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-5 1391  ax-gen 1393  ax-4 1455  ax-17 1474  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-cleq 2093
This theorem is referenced by:  fndmu  5160  fodmrnu  5289  funcoeqres  5332  fvunsng  5546  prarloclem5  7209  addlocprlemeq  7242  zdiv  8991  resqrexlemnm  10630  dvdsmulc  11316  cncongrcoprm  11580
  Copyright terms: Public domain W3C validator