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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 1616 | . . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
4 | 1, 2 | eubid 2033 | . . 3 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) |
5 | 3, 4 | imbi12d 234 | . 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∃!𝑥𝜓) ↔ (∃𝑥𝜒 → ∃!𝑥𝜒))) |
6 | df-mo 2030 | . 2 ⊢ (∃*𝑥𝜓 ↔ (∃𝑥𝜓 → ∃!𝑥𝜓)) | |
7 | df-mo 2030 | . 2 ⊢ (∃*𝑥𝜒 ↔ (∃𝑥𝜒 → ∃!𝑥𝜒)) | |
8 | 5, 6, 7 | 3bitr4g 223 | 1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 Ⅎwnf 1460 ∃wex 1492 ∃!weu 2026 ∃*wmo 2027 |
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-nf 1461 df-eu 2029 df-mo 2030 |
This theorem is referenced by: mobidv 2062 rmobida 2663 rmoeq1f 2671 |
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