Metamath Proof Explorer


Theorem eelT00

Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses eelT00.1 ( ⊤ → 𝜑 )
eelT00.2 𝜓
eelT00.3 𝜒
eelT00.4 ( ( 𝜑𝜓𝜒 ) → 𝜃 )
Assertion eelT00 𝜃

Proof

Step Hyp Ref Expression
1 eelT00.1 ( ⊤ → 𝜑 )
2 eelT00.2 𝜓
3 eelT00.3 𝜒
4 eelT00.4 ( ( 𝜑𝜓𝜒 ) → 𝜃 )
5 3anass ( ( ⊤ ∧ 𝜓𝜒 ) ↔ ( ⊤ ∧ ( 𝜓𝜒 ) ) )
6 truan ( ( ⊤ ∧ ( 𝜓𝜒 ) ) ↔ ( 𝜓𝜒 ) )
7 5 6 bitri ( ( ⊤ ∧ 𝜓𝜒 ) ↔ ( 𝜓𝜒 ) )
8 1 4 syl3an1 ( ( ⊤ ∧ 𝜓𝜒 ) → 𝜃 )
9 7 8 sylbir ( ( 𝜓𝜒 ) → 𝜃 )
10 2 9 mpan ( 𝜒𝜃 )
11 3 10 ax-mp 𝜃