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Theorem 3eqtr4a 2098
Description: A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
3eqtr4a.1 A = B
3eqtr4a.2 (φ𝐶 = A)
3eqtr4a.3 (φ𝐷 = B)
Assertion
Ref Expression
3eqtr4a (φ𝐶 = 𝐷)

Proof of Theorem 3eqtr4a
StepHypRef Expression
1 3eqtr4a.2 . . 3 (φ𝐶 = A)
2 3eqtr4a.1 . . 3 A = B
31, 2syl6eq 2088 . 2 (φ𝐶 = B)
4 3eqtr4a.3 . 2 (φ𝐷 = B)
53, 4eqtr4d 2075 1 (φ𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243
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 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033
This theorem is referenced by:  uniintsnr  3645  fndmdifcom  5219  offres  5707  1stval2  5727  2ndval2  5728  ecovcom  6153  ecovass  6155  ecovdi  6157  zeo  8211  xnegneg  8608  fzsuc2  8803  expnegap0  9010
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