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Theorem syl6eqelr 2442
 Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl6eqelr.1 (φB = A)
syl6eqelr.2 B C
Assertion
Ref Expression
syl6eqelr (φA C)

Proof of Theorem syl6eqelr
StepHypRef Expression
1 syl6eqelr.1 . . 3 (φB = A)
21eqcomd 2358 . 2 (φA = B)
3 syl6eqelr.2 . 2 B C
42, 3syl6eqel 2441 1 (φA C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349 This theorem is referenced by:  cnvkexg  4286  p6exg  4290  ssetkex  4294  sikexg  4296  ins2kexg  4305  ins3kexg  4306  0cnelphi  4597  mapprc  6004  enmap1lem5  6073  nenpw1pwlem2  6085  sbthlem3  6205
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