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Theorem trelded 41054
Description: Deduction form of trel 5155. 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 5155 . . 3 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
543impib 1112 . 2 ((Tr 𝐴𝐵𝐶𝐶𝐴) → 𝐵𝐴)
61, 2, 3, 5syl3an 1156 1 ((𝜑𝜓𝜒) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1083  wcel 2114  Tr wtr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3475  df-in 3920  df-ss 3930  df-uni 4815  df-tr 5149
This theorem is referenced by:  suctrALT3  41413
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