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Theorem eelT01 39455
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelT01.1 (⊤ → 𝜑)
eelT01.2 𝜓
eelT01.3 (𝜒𝜃)
eelT01.4 ((𝜑𝜓𝜃) → 𝜏)
Assertion
Ref Expression
eelT01 (𝜒𝜏)

Proof of Theorem eelT01
StepHypRef Expression
1 3anass 1079 . . 3 ((⊤ ∧ 𝜓𝜒) ↔ (⊤ ∧ (𝜓𝜒)))
2 truan 1648 . . 3 ((⊤ ∧ (𝜓𝜒)) ↔ (𝜓𝜒))
3 simpr 471 . . . 4 ((𝜓𝜒) → 𝜒)
4 eelT01.2 . . . . 5 𝜓
54jctl 507 . . . 4 (𝜒 → (𝜓𝜒))
63, 5impbii 199 . . 3 ((𝜓𝜒) ↔ 𝜒)
71, 2, 63bitri 286 . 2 ((⊤ ∧ 𝜓𝜒) ↔ 𝜒)
8 eelT01.3 . . 3 (𝜒𝜃)
9 eelT01.1 . . . 4 (⊤ → 𝜑)
10 eelT01.4 . . . 4 ((𝜑𝜓𝜃) → 𝜏)
119, 10syl3an1 1165 . . 3 ((⊤ ∧ 𝜓𝜃) → 𝜏)
128, 11syl3an3 1168 . 2 ((⊤ ∧ 𝜓𝜒) → 𝜏)
137, 12sylbir 225 1 (𝜒𝜏)
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