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| Mirrors > Home > MPE Home > Th. List > eubid | Structured version Visualization version GIF version | ||
| Description: Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 19-Feb-2023.) | 
| Ref | Expression | 
|---|---|
| eubid.1 | ⊢ Ⅎ𝑥𝜑 | 
| eubid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| eubid | ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eubid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | eubid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 3 | 1, 2 | alrimi 2213 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) | 
| 4 | eubi 2584 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | |
| 5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 Ⅎwnf 1783 ∃!weu 2568 | 
| 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 ax-6 1967 ax-7 2007 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: euor 2611 euor2 2613 euan 2621 reubida 3407 eusv2i 5394 reusv2lem3 5400 eubiOLD 44455 | 
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