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Mirrors > Home > ILE Home > Th. List > mobid | GIF version |
Description: Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
Ref | Expression |
---|---|
mobid.1 | ⊢ Ⅎ𝑥𝜑 |
mobid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
mobid | ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mobid.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | mobid.2 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | exbid 1609 | . . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
4 | 1, 2 | eubid 2026 | . . 3 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) |
5 | 3, 4 | imbi12d 233 | . 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∃!𝑥𝜓) ↔ (∃𝑥𝜒 → ∃!𝑥𝜒))) |
6 | df-mo 2023 | . 2 ⊢ (∃*𝑥𝜓 ↔ (∃𝑥𝜓 → ∃!𝑥𝜓)) | |
7 | df-mo 2023 | . 2 ⊢ (∃*𝑥𝜒 ↔ (∃𝑥𝜒 → ∃!𝑥𝜒)) | |
8 | 5, 6, 7 | 3bitr4g 222 | 1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 Ⅎwnf 1453 ∃wex 1485 ∃!weu 2019 ∃*wmo 2020 |
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 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-eu 2022 df-mo 2023 |
This theorem is referenced by: mobidv 2055 rmobida 2656 rmoeq1f 2664 |
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