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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 2211 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | eubi 2582 | . 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 1535 Ⅎwnf 1780 ∃!weu 2566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-mo 2538 df-eu 2567 |
This theorem is referenced by: euor 2609 euor2 2611 euan 2619 reubida 3405 eusv2i 5400 reusv2lem3 5406 eubiOLD 44432 |
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