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Theorem sylan9req 2193
 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 2145 . 2 (𝜑𝐴 = 𝐵)
3 sylan9req.2 . 2 (𝜓𝐵 = 𝐶)
42, 3sylan9eq 2192 1 ((𝜑𝜓) → 𝐴 = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331 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 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-cleq 2132 This theorem is referenced by:  fndmu  5224  fodmrnu  5353  funcoeqres  5398  fvunsng  5614  prarloclem5  7308  addlocprlemeq  7341  zdiv  9139  resqrexlemnm  10790  dvdsmulc  11521  cncongrcoprm  11787
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