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Mirrors > Home > ILE Home > Th. List > reupick2 | GIF version |
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
Ref | Expression |
---|---|
reupick2 | ⊢ (((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) ∧ x ∈ A) → (φ ↔ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancr 304 | . . . . . 6 ⊢ ((ψ → φ) → (ψ → (φ ∧ ψ))) | |
2 | 1 | ralimi 2378 | . . . . 5 ⊢ (∀x ∈ A (ψ → φ) → ∀x ∈ A (ψ → (φ ∧ ψ))) |
3 | rexim 2407 | . . . . 5 ⊢ (∀x ∈ A (ψ → (φ ∧ ψ)) → (∃x ∈ A ψ → ∃x ∈ A (φ ∧ ψ))) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (∀x ∈ A (ψ → φ) → (∃x ∈ A ψ → ∃x ∈ A (φ ∧ ψ))) |
5 | reupick3 3216 | . . . . . 6 ⊢ ((∃!x ∈ A φ ∧ ∃x ∈ A (φ ∧ ψ) ∧ x ∈ A) → (φ → ψ)) | |
6 | 5 | 3exp 1102 | . . . . 5 ⊢ (∃!x ∈ A φ → (∃x ∈ A (φ ∧ ψ) → (x ∈ A → (φ → ψ)))) |
7 | 6 | com12 27 | . . . 4 ⊢ (∃x ∈ A (φ ∧ ψ) → (∃!x ∈ A φ → (x ∈ A → (φ → ψ)))) |
8 | 4, 7 | syl6 29 | . . 3 ⊢ (∀x ∈ A (ψ → φ) → (∃x ∈ A ψ → (∃!x ∈ A φ → (x ∈ A → (φ → ψ))))) |
9 | 8 | 3imp1 1116 | . 2 ⊢ (((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) ∧ x ∈ A) → (φ → ψ)) |
10 | rsp 2363 | . . . 4 ⊢ (∀x ∈ A (ψ → φ) → (x ∈ A → (ψ → φ))) | |
11 | 10 | 3ad2ant1 924 | . . 3 ⊢ ((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) → (x ∈ A → (ψ → φ))) |
12 | 11 | imp 115 | . 2 ⊢ (((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) ∧ x ∈ A) → (ψ → φ)) |
13 | 9, 12 | impbid 120 | 1 ⊢ (((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) ∧ x ∈ A) → (φ ↔ ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 ∈ wcel 1390 ∀wral 2300 ∃wrex 2301 ∃!wreu 2302 |
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 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-ral 2305 df-rex 2306 df-reu 2307 |
This theorem is referenced by: (None) |
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