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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 2306 | . . . 4 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ)) | |
2 | exancom 1496 | . . . 4 ⊢ (∃x(x ∈ A ∧ φ) ↔ ∃x(φ ∧ x ∈ A)) | |
3 | 1, 2 | bitri 173 | . . 3 ⊢ (∃x ∈ A φ ↔ ∃x(φ ∧ x ∈ A)) |
4 | nfmo1 1909 | . . . . . 6 ⊢ Ⅎx∃*xφ | |
5 | nfe1 1382 | . . . . . 6 ⊢ Ⅎx∃x(φ ∧ x ∈ A) | |
6 | 4, 5 | nfan 1454 | . . . . 5 ⊢ Ⅎx(∃*xφ ∧ ∃x(φ ∧ x ∈ A)) |
7 | mopick 1975 | . . . . 5 ⊢ ((∃*xφ ∧ ∃x(φ ∧ x ∈ A)) → (φ → x ∈ A)) | |
8 | 6, 7 | alrimi 1412 | . . . 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 2097 | . . . . . 6 ⊢ (x = B → (x ∈ A ↔ B ∈ A)) | |
12 | 10, 11 | imbi12d 223 | . . . . 5 ⊢ (x = B → ((φ → x ∈ A) ↔ (ψ → B ∈ A))) |
13 | 9, 12 | spcv 2640 | . . . 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 271 | . 2 ⊢ ((∃*xφ ∧ ∃x ∈ A φ) → (ψ → B ∈ A)) |
16 | 15 | ancoms 255 | 1 ⊢ ((∃x ∈ A φ ∧ ∃*xφ) → (ψ → B ∈ A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃*wmo 1898 ∃wrex 2301 Vcvv 2551 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 |
This theorem is referenced by: (None) |
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