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Theorem sylan9req 2243
Description: An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
Hypotheses
Ref Expression
sylan9req.1 (𝜑𝐵 = 𝐴)
sylan9req.2 (𝜓𝐵 = 𝐶)
Assertion
Ref Expression
sylan9req ((𝜑𝜓) → 𝐴 = 𝐶)

Proof of Theorem sylan9req
StepHypRef Expression
1 sylan9req.1 . . 3 (𝜑𝐵 = 𝐴)
21eqcomd 2195 . 2 (𝜑𝐴 = 𝐵)
3 sylan9req.2 . 2 (𝜓𝐵 = 𝐶)
42, 3sylan9eq 2242 1 ((𝜑𝜓) → 𝐴 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-cleq 2182
This theorem is referenced by:  fndmu  5336  fodmrnu  5465  funcoeqres  5511  fvunsng  5731  prarloclem5  7530  addlocprlemeq  7563  zdiv  9372  resqrexlemnm  11062  fprodssdc  11633  dvdsmulc  11861  cncongrcoprm  12141  mgmidmo  12851  lgsmodeq  14924
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