Description: The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006) Remove use of ax-5 . (Revised by Steven Nguyen, 9-May-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | moimi.1 | ⊢ ( 𝜑 → 𝜓 ) | |
| Assertion | moimi | ⊢ ( ∃* 𝑥 𝜓 → ∃* 𝑥 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | moimi.1 | ⊢ ( 𝜑 → 𝜓 ) | |
| 2 | 1 | imim1i | ⊢ ( ( 𝜓 → 𝑥 = 𝑦 ) → ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 3 | 2 | alimi | ⊢ ( ∀ 𝑥 ( 𝜓 → 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 4 | 3 | eximi | ⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜓 → 𝑥 = 𝑦 ) → ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) |
| 5 | df-mo | ⊢ ( ∃* 𝑥 𝜓 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜓 → 𝑥 = 𝑦 ) ) | |
| 6 | df-mo | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) | |
| 7 | 4 5 6 | 3imtr4i | ⊢ ( ∃* 𝑥 𝜓 → ∃* 𝑥 𝜑 ) |