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Mirrors > Home > MPE Home > Th. List > moim | Structured version Visualization version GIF version |
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 1810 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦))) |
3 | 2 | eximdv 1913 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
4 | df-mo 2530 | . 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)) | |
5 | df-mo 2530 | . 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 1532 ∃wex 1774 ∃*wmo 2528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 |
This theorem depends on definitions: df-bi 206 df-ex 1775 df-mo 2530 |
This theorem is referenced by: moimdv 2536 mobi 2537 euimmo 2608 moexexlem 2618 rmoim 3734 rmoimi2 3737 ssrmof 4046 disjss3 5142 funmo 6563 funmoOLD 6564 uptx 23523 taylf 26289 |
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