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Theorem eel11111 39475
Description: Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1488 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)
Hypotheses
Ref Expression
eel11111.1 (𝜑𝜓)
eel11111.2 (𝜑𝜒)
eel11111.3 (𝜑𝜃)
eel11111.4 (𝜑𝜏)
eel11111.5 (𝜑𝜂)
eel11111.6 (((((𝜓𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)
Assertion
Ref Expression
eel11111 (𝜑𝜁)

Proof of Theorem eel11111
StepHypRef Expression
1 eel11111.4 . 2 (𝜑𝜏)
2 eel11111.5 . 2 (𝜑𝜂)
3 eel11111.1 . . 3 (𝜑𝜓)
4 eel11111.2 . . 3 (𝜑𝜒)
5 eel11111.3 . . 3 (𝜑𝜃)
6 eel11111.6 . . . . 5 (((((𝜓𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)
76exp41 421 . . . 4 ((𝜓𝜒) → (𝜃 → (𝜏 → (𝜂𝜁))))
87ex 397 . . 3 (𝜓 → (𝜒 → (𝜃 → (𝜏 → (𝜂𝜁)))))
93, 4, 5, 8syl3c 66 . 2 (𝜑 → (𝜏 → (𝜂𝜁)))
101, 2, 9mp2d 49 1 (𝜑𝜁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382
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 197  df-an 383
This theorem is referenced by: (None)
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