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Mirrors > Home > MPE Home > Th. List > mobi | Structured version Visualization version GIF version |
Description: Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.) |
Ref | Expression |
---|---|
mobi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albiim 1890 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) | |
2 | moim 2602 | . . . 4 ⊢ (∀𝑥(𝜓 → 𝜑) → (∃*𝑥𝜑 → ∃*𝑥𝜓)) | |
3 | moim 2602 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) | |
4 | 2, 3 | impbid21d 214 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜑) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))) |
5 | 4 | imp 410 | . 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓)) |
6 | 1, 5 | sylbi 220 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 ∃*wmo 2596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-mo 2598 |
This theorem is referenced by: mobiiOLD 2607 mobidv 2608 mobid 2609 eubi 2644 |
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