Metamath Proof Explorer
Description: A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004)
|
|
Ref |
Expression |
|
Hypotheses |
sylancbr.1 |
⊢ ( 𝜓 ↔ 𝜑 ) |
|
|
sylancbr.2 |
⊢ ( 𝜒 ↔ 𝜑 ) |
|
|
sylancbr.3 |
⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) |
|
Assertion |
sylancbr |
⊢ ( 𝜑 → 𝜃 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
sylancbr.1 |
⊢ ( 𝜓 ↔ 𝜑 ) |
2 |
|
sylancbr.2 |
⊢ ( 𝜒 ↔ 𝜑 ) |
3 |
|
sylancbr.3 |
⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) |
4 |
1 2 3
|
syl2anbr |
⊢ ( ( 𝜑 ∧ 𝜑 ) → 𝜃 ) |
5 |
4
|
anidms |
⊢ ( 𝜑 → 𝜃 ) |