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| Description: The at-most-one quantifier reverses implication. (Contributed by NM, 22-Apr-1995.) | 
| Ref | Expression | 
|---|---|
| moim | ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | imim1 83 | . . . 4 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))) | |
| 2 | 1 | al2imi 1815 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦))) | 
| 3 | 2 | eximdv 1917 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | 
| 4 | df-mo 2540 | . 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)) | |
| 5 | df-mo 2540 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
| 6 | 3, 4, 5 | 3imtr4g 296 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 ∃*wmo 2538 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-mo 2540 | 
| This theorem is referenced by: moimdv 2546 mobi 2547 euimmo 2616 moexexlem 2626 rmoim 3746 rmoimi2 3749 ssrmof 4051 disjss3 5142 funmo 6581 funmoOLD 6582 uptx 23633 taylf 26402 | 
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