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Metamath Proof Explorer
Description: e31an without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011) (Proof modification is discouraged.)
(New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
ee31an.1 |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) |
|
|
ee31an.2 |
⊢ ( 𝜑 → 𝜏 ) |
|
|
ee31an.3 |
⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 ) |
|
Assertion |
ee31an |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ee31an.1 |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) |
| 2 |
|
ee31an.2 |
⊢ ( 𝜑 → 𝜏 ) |
| 3 |
|
ee31an.3 |
⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 ) |
| 4 |
2
|
a1d |
⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) ) |
| 5 |
4
|
a1d |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) ) |
| 6 |
1 5 3
|
ee33an |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) |