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Theorem 3eqtr3g 2092
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 2083 . 2 (φ𝐶 = B)
4 3eqtr3g.3 . 2 B = 𝐷
53, 4syl6eq 2085 1 (φ𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030
This theorem is referenced by:  csbnest1g  2895  dfopg  3538  cores2  4776  funcoeqres  5100  dftpos2  5817  ine0  7167
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