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Theorem eelTT1 41199
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelTT1.1 (⊤ → 𝜑)
eelTT1.2 (⊤ → 𝜓)
eelTT1.3 (𝜒𝜃)
eelTT1.4 ((𝜑𝜓𝜃) → 𝜏)
Assertion
Ref Expression
eelTT1 (𝜒𝜏)

Proof of Theorem eelTT1
StepHypRef Expression
1 3anass 1091 . . 3 ((⊤ ∧ ⊤ ∧ 𝜒) ↔ (⊤ ∧ (⊤ ∧ 𝜒)))
2 anabs5 661 . . 3 ((⊤ ∧ (⊤ ∧ 𝜒)) ↔ (⊤ ∧ 𝜒))
3 truan 1548 . . 3 ((⊤ ∧ 𝜒) ↔ 𝜒)
41, 2, 33bitri 299 . 2 ((⊤ ∧ ⊤ ∧ 𝜒) ↔ 𝜒)
5 eelTT1.3 . . 3 (𝜒𝜃)
6 eelTT1.2 . . . 4 (⊤ → 𝜓)
7 eelTT1.1 . . . . 5 (⊤ → 𝜑)
8 eelTT1.4 . . . . 5 ((𝜑𝜓𝜃) → 𝜏)
97, 8syl3an1 1159 . . . 4 ((⊤ ∧ 𝜓𝜃) → 𝜏)
106, 9syl3an2 1160 . . 3 ((⊤ ∧ ⊤ ∧ 𝜃) → 𝜏)
115, 10syl3an3 1161 . 2 ((⊤ ∧ ⊤ ∧ 𝜒) → 𝜏)
124, 11sylbir 237 1 (𝜒𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083  wtru 1538
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 209  df-an 399  df-3an 1085  df-tru 1540
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator