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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  6438  nq0m0r  6552  genpelxp  6607  un0mulcl  8214  znegcl  8274  zeo  8341  eqreznegel  8547  xnegcl  8743  iser0  9224  expival  9231  expcllem  9240  m1expcl2  9251  resqrexlemlo  9585  iserige0  9837
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