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 |
eelT01.1 |
⊢ ( ⊤ → 𝜑 ) |
|
|
eelT01.2 |
⊢ 𝜓 |
|
|
eelT01.3 |
⊢ ( 𝜒 → 𝜃 ) |
|
|
eelT01.4 |
⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 ) |
|
Assertion |
eelT01 |
⊢ ( 𝜒 → 𝜏 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
eelT01.1 |
⊢ ( ⊤ → 𝜑 ) |
2 |
|
eelT01.2 |
⊢ 𝜓 |
3 |
|
eelT01.3 |
⊢ ( 𝜒 → 𝜃 ) |
4 |
|
eelT01.4 |
⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 ) |
5 |
|
3anass |
⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜒 ) ↔ ( ⊤ ∧ ( 𝜓 ∧ 𝜒 ) ) ) |
6 |
|
truan |
⊢ ( ( ⊤ ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( 𝜓 ∧ 𝜒 ) ) |
7 |
|
simpr |
⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜒 ) |
8 |
2
|
jctl |
⊢ ( 𝜒 → ( 𝜓 ∧ 𝜒 ) ) |
9 |
7 8
|
impbii |
⊢ ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜒 ) |
10 |
5 6 9
|
3bitri |
⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜒 ) ↔ 𝜒 ) |
11 |
1 4
|
syl3an1 |
⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜃 ) → 𝜏 ) |
12 |
3 11
|
syl3an3 |
⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜒 ) → 𝜏 ) |
13 |
10 12
|
sylbir |
⊢ ( 𝜒 → 𝜏 ) |