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Theorem sylan9req 2141
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 2093 . 2 (𝜑𝐴 = 𝐵)
3 sylan9req.2 . 2 (𝜓𝐵 = 𝐶)
42, 3sylan9eq 2140 1 ((𝜑𝜓) → 𝐴 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081
This theorem is referenced by:  xpid11m  4658  fndmu  5115  fodmrnu  5241  funcoeqres  5284  fvunsng  5491  prarloclem5  7057  addlocprlemeq  7090  zdiv  8832  resqrexlemnm  10447  dvdsmulc  11098  cncongrcoprm  11362
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