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Theorem 3eqtr3a 2093
Description: A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
Hypotheses
Ref Expression
3eqtr3a.1 A = B
3eqtr3a.2 (φA = 𝐶)
3eqtr3a.3 (φB = 𝐷)
Assertion
Ref Expression
3eqtr3a (φ𝐶 = 𝐷)

Proof of Theorem 3eqtr3a
StepHypRef Expression
1 3eqtr3a.2 . 2 (φA = 𝐶)
2 3eqtr3a.1 . . 3 A = B
3 3eqtr3a.3 . . 3 (φB = 𝐷)
42, 3syl5eq 2081 . 2 (φA = 𝐷)
51, 4eqtr3d 2071 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:  uneqin  3182  coi2  4780  foima  5054  f1imacnv  5086  fvsnun2  5304  halfnqq  6393
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