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Metamath Proof Explorer
Description: Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007) (Proof shortened by Wolf Lammen, 19-Jul-2013)
|
|
Ref |
Expression |
|
Hypotheses |
anim12ii.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
|
|
anim12ii.2 |
⊢ ( 𝜃 → ( 𝜓 → 𝜏 ) ) |
|
Assertion |
anim12ii |
⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 → ( 𝜒 ∧ 𝜏 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
anim12ii.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
| 2 |
|
anim12ii.2 |
⊢ ( 𝜃 → ( 𝜓 → 𝜏 ) ) |
| 3 |
|
pm3.43 |
⊢ ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜓 → 𝜏 ) ) → ( 𝜓 → ( 𝜒 ∧ 𝜏 ) ) ) |
| 4 |
1 2 3
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 → ( 𝜒 ∧ 𝜏 ) ) ) |