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Theorem eelT11 38758
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelT11.1 (⊤ → 𝜑)
eelT11.2 (𝜓𝜒)
eelT11.3 (𝜓𝜃)
eelT11.4 ((𝜑𝜒𝜃) → 𝜏)
Assertion
Ref Expression
eelT11 (𝜓𝜏)

Proof of Theorem eelT11
StepHypRef Expression
1 3anass 1041 . . 3 ((⊤ ∧ 𝜓𝜓) ↔ (⊤ ∧ (𝜓𝜓)))
2 truan 1500 . . 3 ((⊤ ∧ (𝜓𝜓)) ↔ (𝜓𝜓))
3 anidm 676 . . 3 ((𝜓𝜓) ↔ 𝜓)
41, 2, 33bitri 286 . 2 ((⊤ ∧ 𝜓𝜓) ↔ 𝜓)
5 eelT11.3 . . 3 (𝜓𝜃)
6 eelT11.2 . . . 4 (𝜓𝜒)
7 eelT11.1 . . . . 5 (⊤ → 𝜑)
8 eelT11.4 . . . . 5 ((𝜑𝜒𝜃) → 𝜏)
97, 8syl3an1 1358 . . . 4 ((⊤ ∧ 𝜒𝜃) → 𝜏)
106, 9syl3an2 1359 . . 3 ((⊤ ∧ 𝜓𝜃) → 𝜏)
115, 10syl3an3 1360 . 2 ((⊤ ∧ 𝜓𝜓) → 𝜏)
124, 11sylbir 225 1 (𝜓𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037  wtru 1483
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 1039  df-tru 1485
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator