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Theorem eelTTT 41830
 Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelTTT.1 (⊤ → 𝜑)
eelTTT.2 (⊤ → 𝜓)
eelTTT.3 (⊤ → 𝜒)
eelTTT.4 ((𝜑𝜓𝜒) → 𝜃)
Assertion
Ref Expression
eelTTT 𝜃

Proof of Theorem eelTTT
StepHypRef Expression
1 eelTTT.3 . . 3 (⊤ → 𝜒)
2 truan 1549 . . . 4 ((⊤ ∧ 𝜒) ↔ 𝜒)
3 eelTTT.2 . . . . 5 (⊤ → 𝜓)
4 3anass 1092 . . . . . . 7 ((⊤ ∧ 𝜓𝜒) ↔ (⊤ ∧ (𝜓𝜒)))
5 truan 1549 . . . . . . 7 ((⊤ ∧ (𝜓𝜒)) ↔ (𝜓𝜒))
64, 5bitri 278 . . . . . 6 ((⊤ ∧ 𝜓𝜒) ↔ (𝜓𝜒))
7 eelTTT.1 . . . . . . 7 (⊤ → 𝜑)
8 eelTTT.4 . . . . . . 7 ((𝜑𝜓𝜒) → 𝜃)
97, 8syl3an1 1160 . . . . . 6 ((⊤ ∧ 𝜓𝜒) → 𝜃)
106, 9sylbir 238 . . . . 5 ((𝜓𝜒) → 𝜃)
113, 10sylan 583 . . . 4 ((⊤ ∧ 𝜒) → 𝜃)
122, 11sylbir 238 . . 3 (𝜒𝜃)
131, 12syl 17 . 2 (⊤ → 𝜃)
1413mptru 1545 1 𝜃
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084  ⊤wtru 1539 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 210  df-an 400  df-3an 1086  df-tru 1541 This theorem is referenced by: (None)
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