Description: Theorem 19.22 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994) (Proof shortened by Andrew Salmon, 30-May-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | rexim | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | con3 | ⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜑 ) ) | |
| 2 | 1 | ral2imi | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 → ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ) ) |
| 3 | ralnex | ⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜓 ) | |
| 4 | ralnex | ⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜑 ) | |
| 5 | 2 3 4 | 3imtr3g | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ¬ ∃ 𝑥 ∈ 𝐴 𝜓 → ¬ ∃ 𝑥 ∈ 𝐴 𝜑 ) ) |
| 6 | 5 | con4d | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) ) |