Description: e333 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ee333.1 | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) | |
ee333.2 | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) ) | ||
ee333.3 | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) | ||
ee333.4 | ⊢ ( 𝜃 → ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) | ||
Assertion | ee333 | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜁 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ee333.1 | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) | |
2 | ee333.2 | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) ) | |
3 | ee333.3 | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) | |
4 | ee333.4 | ⊢ ( 𝜃 → ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) | |
5 | 1 | 3imp | ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) |
6 | 2 | 3imp | ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜏 ) |
7 | 3 | 3imp | ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜂 ) |
8 | 5 6 7 4 | syl3c | ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜁 ) |
9 | 8 | 3exp | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜁 ) ) ) |