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