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Metamath Proof Explorer
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004) (Proof shortened by Wolf Lammen, 4-Dec-2012)
|
|
Ref |
Expression |
|
Hypothesis |
adantr2.1 |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) → 𝜃 ) |
|
Assertion |
adantrll |
⊢ ( ( 𝜑 ∧ ( ( 𝜏 ∧ 𝜓 ) ∧ 𝜒 ) ) → 𝜃 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
adantr2.1 |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) → 𝜃 ) |
| 2 |
|
simpr |
⊢ ( ( 𝜏 ∧ 𝜓 ) → 𝜓 ) |
| 3 |
2 1
|
sylanr1 |
⊢ ( ( 𝜑 ∧ ( ( 𝜏 ∧ 𝜓 ) ∧ 𝜒 ) ) → 𝜃 ) |