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| Mirrors > Home > MPE Home > Th. List > morex | Structured version Visualization version GIF version | ||
| Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| Ref | Expression |
|---|---|
| morex.1 | ⊢ 𝐵 ∈ V |
| morex.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| morex | ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-rex 3071 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | exancom 1861 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) | |
| 3 | 1, 2 | bitri 275 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) |
| 4 | nfmo1 2557 | . . . . . 6 ⊢ Ⅎ𝑥∃*𝑥𝜑 | |
| 5 | nfe1 2150 | . . . . . 6 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴) | |
| 6 | 4, 5 | nfan 1899 | . . . . 5 ⊢ Ⅎ𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) |
| 7 | mopick 2625 | . . . . 5 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜑 → 𝑥 ∈ 𝐴)) | |
| 8 | 6, 7 | alrimi 2213 | . . . 4 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) |
| 9 | morex.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
| 10 | morex.2 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
| 11 | eleq1 2829 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
| 12 | 10, 11 | imbi12d 344 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜓 → 𝐵 ∈ 𝐴))) |
| 13 | 9, 12 | spcv 3605 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) → (𝜓 → 𝐵 ∈ 𝐴)) |
| 14 | 8, 13 | syl 17 | . . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜓 → 𝐵 ∈ 𝐴)) |
| 15 | 3, 14 | sylan2b 594 | . 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) |
| 16 | 15 | ancoms 458 | 1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃*wmo 2538 ∃wrex 3070 Vcvv 3480 |
| 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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 |
| This theorem is referenced by: (None) |
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