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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.) |
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 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | exbid 2258 | . . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
4 | 1, 2 | mobid 2607 | . . 3 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
5 | 3, 4 | anbi12d 625 | . 2 ⊢ (𝜑 → ((∃𝑥𝜓 ∧ ∃*𝑥𝜓) ↔ (∃𝑥𝜒 ∧ ∃*𝑥𝜒))) |
6 | df-eu 2609 | . 2 ⊢ (∃!𝑥𝜓 ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓)) | |
7 | df-eu 2609 | . 2 ⊢ (∃!𝑥𝜒 ↔ (∃𝑥𝜒 ∧ ∃*𝑥𝜒)) | |
8 | 5, 6, 7 | 3bitr4g 306 | 1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∃wex 1875 Ⅎwnf 1879 ∃*wmo 2589 ∃!weu 2608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-12 2213 |
This theorem depends on definitions: df-bi 199 df-an 386 df-ex 1876 df-nf 1880 df-mo 2591 df-eu 2609 |
This theorem is referenced by: eubidvOLDOLD 2643 mobidOLD 2646 euor 2663 euor2 2666 euan 2684 reubida 3305 reueq1f 3318 eusv2i 5063 reusv2lem3 5069 eubiOLD 39407 |
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