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Theorem eel0T1 38458
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eel0T1.1 𝜑
eel0T1.2 (⊤ → 𝜓)
eel0T1.3 (𝜒𝜃)
eel0T1.4 ((𝜑𝜓𝜃) → 𝜏)
Assertion
Ref Expression
eel0T1 (𝜒𝜏)

Proof of Theorem eel0T1
StepHypRef Expression
1 3anass 1040 . . 3 ((𝜑 ∧ ⊤ ∧ 𝜒) ↔ (𝜑 ∧ (⊤ ∧ 𝜒)))
2 simpr 477 . . . 4 ((𝜑 ∧ (⊤ ∧ 𝜒)) → (⊤ ∧ 𝜒))
3 eel0T1.1 . . . . 5 𝜑
43jctl 563 . . . 4 ((⊤ ∧ 𝜒) → (𝜑 ∧ (⊤ ∧ 𝜒)))
52, 4impbii 199 . . 3 ((𝜑 ∧ (⊤ ∧ 𝜒)) ↔ (⊤ ∧ 𝜒))
6 truan 1498 . . 3 ((⊤ ∧ 𝜒) ↔ 𝜒)
71, 5, 63bitri 286 . 2 ((𝜑 ∧ ⊤ ∧ 𝜒) ↔ 𝜒)
8 eel0T1.3 . . 3 (𝜒𝜃)
9 eel0T1.2 . . . 4 (⊤ → 𝜓)
10 eel0T1.4 . . . 4 ((𝜑𝜓𝜃) → 𝜏)
119, 10syl3an2 1357 . . 3 ((𝜑 ∧ ⊤ ∧ 𝜃) → 𝜏)
128, 11syl3an3 1358 . 2 ((𝜑 ∧ ⊤ ∧ 𝜒) → 𝜏)
137, 12sylbir 225 1 (𝜒𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wtru 1481
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 386  df-3an 1038  df-tru 1483
This theorem is referenced by: (None)
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