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Theorem sylan9req 2194
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 2146 . 2  |-  ( ph  ->  A  =  B )
3 sylan9req.2 . 2  |-  ( ps 
->  B  =  C
)
42, 3sylan9eq 2193 1  |-  ( (
ph  /\  ps )  ->  A  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332
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-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-cleq 2133
This theorem is referenced by:  fndmu  5228  fodmrnu  5357  funcoeqres  5402  fvunsng  5618  prarloclem5  7328  addlocprlemeq  7361  zdiv  9159  resqrexlemnm  10818  dvdsmulc  11548  cncongrcoprm  11814
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