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Theorem eelT00 39449
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eelT00.1 (⊤ → 𝜑)
eelT00.2 𝜓
eelT00.3 𝜒
eelT00.4 ((𝜑𝜓𝜒) → 𝜃)
Assertion
Ref Expression
eelT00 𝜃

Proof of Theorem eelT00
StepHypRef Expression
1 eelT00.3 . 2 𝜒
2 eelT00.2 . . 3 𝜓
3 3anass 1079 . . . . 5 ((⊤ ∧ 𝜓𝜒) ↔ (⊤ ∧ (𝜓𝜒)))
4 truan 1648 . . . . 5 ((⊤ ∧ (𝜓𝜒)) ↔ (𝜓𝜒))
53, 4bitri 264 . . . 4 ((⊤ ∧ 𝜓𝜒) ↔ (𝜓𝜒))
6 eelT00.1 . . . . 5 (⊤ → 𝜑)
7 eelT00.4 . . . . 5 ((𝜑𝜓𝜒) → 𝜃)
86, 7syl3an1 1165 . . . 4 ((⊤ ∧ 𝜓𝜒) → 𝜃)
95, 8sylbir 225 . . 3 ((𝜓𝜒) → 𝜃)
102, 9mpan 662 . 2 (𝜒𝜃)
111, 10ax-mp 5 1 𝜃
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)
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