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Mirrors > Home > ILE Home > Th. List > morex | 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 2441 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | exancom 1588 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | bitri 183 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) |
4 | nfmo1 2018 | . . . . . 6 ⊢ Ⅎ𝑥∃*𝑥𝜑 | |
5 | nfe1 1476 | . . . . . 6 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴) | |
6 | 4, 5 | nfan 1545 | . . . . 5 ⊢ Ⅎ𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) |
7 | mopick 2084 | . . . . 5 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜑 → 𝑥 ∈ 𝐴)) | |
8 | 6, 7 | alrimi 1502 | . . . 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 2220 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
12 | 10, 11 | imbi12d 233 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜓 → 𝐵 ∈ 𝐴))) |
13 | 9, 12 | spcv 2806 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) → (𝜓 → 𝐵 ∈ 𝐴)) |
14 | 8, 13 | syl 14 | . . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜓 → 𝐵 ∈ 𝐴)) |
15 | 3, 14 | sylan2b 285 | . 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) |
16 | 15 | ancoms 266 | 1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1333 = wceq 1335 ∃wex 1472 ∃*wmo 2007 ∈ wcel 2128 ∃wrex 2436 Vcvv 2712 |
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-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 |
This theorem is referenced by: (None) |
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