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Theorem trelded 41906
Description: Deduction form of trel 5185. 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 5185 . . 3 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
543impib 1118 . 2 ((Tr 𝐴𝐵𝐶𝐶𝐴) → 𝐵𝐴)
61, 2, 3, 5syl3an 1162 1 ((𝜑𝜓𝜒) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089  wcel 2112  Tr wtr 5178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-v 3425  df-in 3890  df-ss 3900  df-uni 4837  df-tr 5179
This theorem is referenced by:  suctrALT3  42265
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