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Theorem 3eqtr3g 2143
Description: A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
Hypotheses
Ref Expression
3eqtr3g.1 (𝜑𝐴 = 𝐵)
3eqtr3g.2 𝐴 = 𝐶
3eqtr3g.3 𝐵 = 𝐷
Assertion
Ref Expression
3eqtr3g (𝜑𝐶 = 𝐷)

Proof of Theorem 3eqtr3g
StepHypRef Expression
1 3eqtr3g.2 . . 3 𝐴 = 𝐶
2 3eqtr3g.1 . . 3 (𝜑𝐴 = 𝐵)
31, 2syl5eqr 2134 . 2 (𝜑𝐶 = 𝐵)
4 3eqtr3g.3 . 2 𝐵 = 𝐷
53, 4syl6eq 2136 1 (𝜑𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = 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:  csbnest1g  2981  dfopg  3615  cores2  4930  funcoeqres  5268  dftpos2  6008  ine0  7851  fisumcom2  10795  fisum0diag2  10804
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