Description: Theorem 19.26 of Margaris p. 90. Also Theorem *10.22 of WhiteheadRussell p. 147. (Contributed by NM, 12-Mar-1993) (Proof shortened by Wolf Lammen, 4-Jul-2014)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 19.26 | ⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl | ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) | |
| 2 | 1 | alimi | ⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 𝜑 ) |
| 3 | simpr | ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜓 ) | |
| 4 | 3 | alimi | ⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 𝜓 ) |
| 5 | 2 4 | jca | ⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) ) |
| 6 | id | ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) | |
| 7 | 6 | alanimi | ⊢ ( ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ) |
| 8 | 5 7 | impbii | ⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) ) |