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Theorem sylan9req 2109
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 2061 . 2  |-  ( ph  ->  A  =  B )
3 sylan9req.2 . 2  |-  ( ps 
->  B  =  C
)
42, 3sylan9eq 2108 1  |-  ( (
ph  /\  ps )  ->  A  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049
This theorem is referenced by:  xpid11m  4585  fndmu  5028  fodmrnu  5142  funcoeqres  5185  fvunsng  5385  prarloclem5  6656  addlocprlemeq  6689  zdiv  8386  resqrexlemnm  9845  dvdsmulc  10135
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