ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sylan9req GIF version

Theorem sylan9req 2109
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 2061 . 2 (𝜑𝐴 = 𝐵)
3 sylan9req.2 . 2 (𝜓𝐵 = 𝐶)
42, 3sylan9eq 2108 1 ((𝜑𝜓) → 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  4584  fndmu  5027  fodmrnu  5141  funcoeqres  5184  fvunsng  5384  prarloclem5  6655  addlocprlemeq  6688  zdiv  8385  resqrexlemnm  9844  dvdsmulc  10134
  Copyright terms: Public domain W3C validator