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Theorem trelded 39604
Description: Deduction form of trel 4984. 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 4984 . . 3 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
543impib 1148 . 2 ((Tr 𝐴𝐵𝐶𝐶𝐴) → 𝐵𝐴)
61, 2, 3, 5syl3an 1203 1 ((𝜑𝜓𝜒) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1111  wcel 2164  Tr wtr 4977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-in 3805  df-ss 3812  df-uni 4661  df-tr 4978
This theorem is referenced by:  suctrALT3  39973
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