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Theorem eelT12 39453
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelT12.1 (⊤ → 𝜑)
eelT12.2 (𝜓𝜒)
eelT12.3 (𝜃𝜏)
eelT12.4 ((𝜑𝜒𝜏) → 𝜂)
Assertion
Ref Expression
eelT12 ((𝜓𝜃) → 𝜂)

Proof of Theorem eelT12
StepHypRef Expression
1 3anass 1079 . . 3 ((⊤ ∧ 𝜓𝜃) ↔ (⊤ ∧ (𝜓𝜃)))
2 truan 1648 . . 3 ((⊤ ∧ (𝜓𝜃)) ↔ (𝜓𝜃))
31, 2bitri 264 . 2 ((⊤ ∧ 𝜓𝜃) ↔ (𝜓𝜃))
4 eelT12.3 . . 3 (𝜃𝜏)
5 eelT12.2 . . . 4 (𝜓𝜒)
6 eelT12.1 . . . . 5 (⊤ → 𝜑)
7 eelT12.4 . . . . 5 ((𝜑𝜒𝜏) → 𝜂)
86, 7syl3an1 1165 . . . 4 ((⊤ ∧ 𝜒𝜏) → 𝜂)
95, 8syl3an2 1166 . . 3 ((⊤ ∧ 𝜓𝜏) → 𝜂)
104, 9syl3an3 1168 . 2 ((⊤ ∧ 𝜓𝜃) → 𝜂)
113, 10sylbir 225 1 ((𝜓𝜃) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070  wtru 1631
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  df-3an 1072  df-tru 1633
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator