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Theorem 3imp3i2an 1270
Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
Hypotheses
Ref Expression
3imp3i2an.1 ((𝜑𝜓𝜒) → 𝜃)
3imp3i2an.2 ((𝜑𝜒) → 𝜏)
3imp3i2an.3 ((𝜃𝜏) → 𝜂)
Assertion
Ref Expression
3imp3i2an ((𝜑𝜓𝜒) → 𝜂)

Proof of Theorem 3imp3i2an
StepHypRef Expression
1 3imp3i2an.2 . . . . . . . 8 ((𝜑𝜒) → 𝜏)
2 3imp3i2an.1 . . . . . . . . . . 11 ((𝜑𝜓𝜒) → 𝜃)
323exp 1256 . . . . . . . . . 10 (𝜑 → (𝜓 → (𝜒𝜃)))
4 3imp3i2an.3 . . . . . . . . . . 11 ((𝜃𝜏) → 𝜂)
54ex 449 . . . . . . . . . 10 (𝜃 → (𝜏𝜂))
63, 5syl8 74 . . . . . . . . 9 (𝜑 → (𝜓 → (𝜒 → (𝜏𝜂))))
76com4r 92 . . . . . . . 8 (𝜏 → (𝜑 → (𝜓 → (𝜒𝜂))))
81, 7syl 17 . . . . . . 7 ((𝜑𝜒) → (𝜑 → (𝜓 → (𝜒𝜂))))
98ex 449 . . . . . 6 (𝜑 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
109pm2.43b 53 . . . . 5 (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))
1110com4r 92 . . . 4 (𝜒 → (𝜒 → (𝜑 → (𝜓𝜂))))
1211pm2.43i 50 . . 3 (𝜒 → (𝜑 → (𝜓𝜂)))
1312com3l 87 . 2 (𝜑 → (𝜓 → (𝜒𝜂)))
14133imp 1249 1 ((𝜑𝜓𝜒) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033
This theorem is referenced by:  upgr2pthnlp  40930  av-frgrareg  41540
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