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Metamath Proof Explorer
Description: Deduction joining nested implications to form implication of
conjunctions. (Contributed by NM, 29-Feb-1996)
|
|
Ref |
Expression |
|
Hypotheses |
im2an9.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
|
|
im2an9.2 |
⊢ ( 𝜃 → ( 𝜏 → 𝜂 ) ) |
|
Assertion |
im2anan9r |
⊢ ( ( 𝜃 ∧ 𝜑 ) → ( ( 𝜓 ∧ 𝜏 ) → ( 𝜒 ∧ 𝜂 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
im2an9.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
| 2 |
|
im2an9.2 |
⊢ ( 𝜃 → ( 𝜏 → 𝜂 ) ) |
| 3 |
1 2
|
im2anan9 |
⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ∧ 𝜏 ) → ( 𝜒 ∧ 𝜂 ) ) ) |
| 4 |
3
|
ancoms |
⊢ ( ( 𝜃 ∧ 𝜑 ) → ( ( 𝜓 ∧ 𝜏 ) → ( 𝜒 ∧ 𝜂 ) ) ) |