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Mirrors > Home > ILE Home > Th. List > eubidh | GIF version |
Description: Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
Ref | Expression |
---|---|
eubidh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
eubidh.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
eubidh | ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eubidh.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | eubidh.2 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | bibi1d 233 | . . . 4 ⊢ (𝜑 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜒 ↔ 𝑥 = 𝑦))) |
4 | 1, 3 | albidh 1480 | . . 3 ⊢ (𝜑 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜒 ↔ 𝑥 = 𝑦))) |
5 | 4 | exbidv 1825 | . 2 ⊢ (𝜑 → (∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦))) |
6 | df-eu 2029 | . 2 ⊢ (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦)) | |
7 | df-eu 2029 | . 2 ⊢ (∃!𝑥𝜒 ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦)) | |
8 | 5, 6, 7 | 3bitr4g 223 | 1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 ∃wex 1492 ∃!weu 2026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-eu 2029 |
This theorem is referenced by: euor 2052 mobidh 2060 euan 2082 euor2 2084 eupickbi 2108 |
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