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Theorem trelded 42185
Description: Deduction form of trel 5198. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
trelded.1 (𝜑 → Tr 𝐴)
trelded.2 (𝜓𝐵𝐶)
trelded.3 (𝜒𝐶𝐴)
Assertion
Ref Expression
trelded ((𝜑𝜓𝜒) → 𝐵𝐴)

Proof of Theorem trelded
StepHypRef Expression
1 trelded.1 . 2 (𝜑 → Tr 𝐴)
2 trelded.2 . 2 (𝜓𝐵𝐶)
3 trelded.3 . 2 (𝜒𝐶𝐴)
4 trel 5198 . . 3 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
543impib 1115 . 2 ((Tr 𝐴𝐵𝐶𝐶𝐴) → 𝐵𝐴)
61, 2, 3, 5syl3an 1159 1 ((𝜑𝜓𝜒) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2106  Tr wtr 5191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-tr 5192
This theorem is referenced by:  suctrALT3  42544
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