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| Mirrors > Home > MPE Home > Th. List > mobid | Structured version Visualization version GIF version | ||
| Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2141, ax-11 2157, ax-13 2377. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.) |
| Ref | Expression |
|---|---|
| mobid.1 | ⊢ Ⅎ𝑥𝜑 |
| mobid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| mobid | ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mobid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | mobid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 3 | 1, 2 | alrimi 2213 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
| 4 | mobi 2547 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | |
| 5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 Ⅎwnf 1783 ∃*wmo 2538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-mo 2540 |
| This theorem is referenced by: moanim 2620 rmobida 3406 rmoeq1f 3424 funcnvmpt 32677 |
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