Metamath Proof Explorer
Description: syl2an with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016)
|
|
Ref |
Expression |
|
Hypotheses |
eel2131.1 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) |
|
|
eel2131.2 |
⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 ) |
|
|
eel2131.3 |
⊢ ( ( 𝜒 ∧ 𝜏 ) → 𝜂 ) |
|
Assertion |
eel2131 |
⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜂 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
eel2131.1 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) |
2 |
|
eel2131.2 |
⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 ) |
3 |
|
eel2131.3 |
⊢ ( ( 𝜒 ∧ 𝜏 ) → 𝜂 ) |
4 |
1 2 3
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜃 ) ) → 𝜂 ) |
5 |
4
|
3impdi |
⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜂 ) |