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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 2210 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | eubi 2586 | . 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 205 ∀wal 1540 Ⅎwnf 1790 ∃!weu 2570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 df-nf 1791 df-mo 2542 df-eu 2571 |
This theorem is referenced by: euor 2615 euor2 2617 euan 2625 reubida 3320 reueq1f 3333 eusv2i 5321 reusv2lem3 5327 eubiOLD 42022 |
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