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Mirrors > Home > MPE Home > Th. List > eubid | Structured version Visualization version GIF version |
Description: Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Ref | Expression |
---|---|
eubid.1 | ⊢ Ⅎ𝑥𝜑 |
eubid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
eubid | ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eubid.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | eubid.2 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | bibi1d 332 | . . . 4 ⊢ (𝜑 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜒 ↔ 𝑥 = 𝑦))) |
4 | 1, 3 | albid 2077 | . . 3 ⊢ (𝜑 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜒 ↔ 𝑥 = 𝑦))) |
5 | 4 | exbidv 1837 | . 2 ⊢ (𝜑 → (∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦))) |
6 | df-eu 2462 | . 2 ⊢ (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦)) | |
7 | df-eu 2462 | . 2 ⊢ (∃!𝑥𝜒 ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦)) | |
8 | 5, 6, 7 | 3bitr4g 302 | 1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 Ⅎwnf 1699 ∃!weu 2458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nf 1701 df-eu 2462 |
This theorem is referenced by: mobid 2477 eubidv 2478 euor 2500 euor2 2502 euan 2518 reubida 3101 reueq1f 3113 eusv2i 4789 reusv2lem3 4797 eubi 37659 nbusgredgeu0 40596 |
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