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Mirrors > Home > NFE 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 2620 | . . . 4 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ)) | |
2 | exancom 1586 | . . . 4 ⊢ (∃x(x ∈ A ∧ φ) ↔ ∃x(φ ∧ x ∈ A)) | |
3 | 1, 2 | bitri 240 | . . 3 ⊢ (∃x ∈ A φ ↔ ∃x(φ ∧ x ∈ A)) |
4 | nfmo1 2215 | . . . . . 6 ⊢ Ⅎx∃*xφ | |
5 | nfe1 1732 | . . . . . 6 ⊢ Ⅎx∃x(φ ∧ x ∈ A) | |
6 | 4, 5 | nfan 1824 | . . . . 5 ⊢ Ⅎx(∃*xφ ∧ ∃x(φ ∧ x ∈ A)) |
7 | mopick 2266 | . . . . 5 ⊢ ((∃*xφ ∧ ∃x(φ ∧ x ∈ A)) → (φ → x ∈ A)) | |
8 | 6, 7 | alrimi 1765 | . . . 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 2413 | . . . . . 6 ⊢ (x = B → (x ∈ A ↔ B ∈ A)) | |
12 | 10, 11 | imbi12d 311 | . . . . 5 ⊢ (x = B → ((φ → x ∈ A) ↔ (ψ → B ∈ A))) |
13 | 9, 12 | spcv 2945 | . . . 4 ⊢ (∀x(φ → x ∈ A) → (ψ → B ∈ A)) |
14 | 8, 13 | syl 15 | . . 3 ⊢ ((∃*xφ ∧ ∃x(φ ∧ x ∈ A)) → (ψ → B ∈ A)) |
15 | 3, 14 | sylan2b 461 | . 2 ⊢ ((∃*xφ ∧ ∃x ∈ A φ) → (ψ → B ∈ A)) |
16 | 15 | ancoms 439 | 1 ⊢ ((∃x ∈ A φ ∧ ∃*xφ) → (ψ → B ∈ A)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃*wmo 2205 ∃wrex 2615 Vcvv 2859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-v 2861 |
This theorem is referenced by: (None) |
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