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Theorem syl6eqel 2128
 Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl6eqel.1 (𝜑𝐴 = 𝐵)
syl6eqel.2 𝐵𝐶
Assertion
Ref Expression
syl6eqel (𝜑𝐴𝐶)

Proof of Theorem syl6eqel
StepHypRef Expression
1 syl6eqel.1 . 2 (𝜑𝐴 = 𝐵)
2 syl6eqel.2 . . 3 𝐵𝐶
32a1i 9 . 2 (𝜑𝐵𝐶)
41, 3eqeltrd 2114 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393 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-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036 This theorem is referenced by:  syl6eqelr  2129  snexprc  3938  onsucelsucexmidlem  4254  ovprc  5540  nnmcl  6060  xpsnen  6295  indpi  6440  nq0m0r  6554  genpelxp  6609  un0mulcl  8216  znegcl  8276  zeo  8343  eqreznegel  8549  xnegcl  8745  iser0  9250  expival  9257  expcllem  9266  m1expcl2  9277  resqrexlemlo  9611  iserige0  9863
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