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Mirrors > Home > ILE Home > Th. List > eubid | GIF version |
Description: Formula-building rule for 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 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | bibi1d 232 | . . . 4 ⊢ (𝜑 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜒 ↔ 𝑥 = 𝑦))) |
4 | 1, 3 | albid 1595 | . . 3 ⊢ (𝜑 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜒 ↔ 𝑥 = 𝑦))) |
5 | 4 | exbidv 1805 | . 2 ⊢ (𝜑 → (∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦))) |
6 | df-eu 2009 | . 2 ⊢ (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦)) | |
7 | df-eu 2009 | . 2 ⊢ (∃!𝑥𝜒 ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦)) | |
8 | 5, 6, 7 | 3bitr4g 222 | 1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1333 Ⅎwnf 1440 ∃wex 1472 ∃!weu 2006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-17 1506 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-eu 2009 |
This theorem is referenced by: eubidv 2014 mobid 2041 reubida 2638 reueq1f 2650 eusv2i 4417 |
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