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Theorem syl6sseq 3056
Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
syl6sseq.1 (𝜑𝐴𝐵)
syl6sseq.2 𝐵 = 𝐶
Assertion
Ref Expression
syl6sseq (𝜑𝐴𝐶)

Proof of Theorem syl6sseq
StepHypRef Expression
1 syl6sseq.1 . 2 (𝜑𝐴𝐵)
2 syl6sseq.2 . . 3 𝐵 = 𝐶
32sseq2i 3035 . 2 (𝐴𝐵𝐴𝐶)
41, 3sylib 120 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2990  df-ss 2997
This theorem is referenced by:  syl6sseqr  3057  onintonm  4297  relrelss  4911  iotanul  4949  foimacnv  5219  cauappcvgprlemladdru  7118
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