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Mirrors > Home > ILE Home > Th. List > eqreu | GIF version |
Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
eqreu.1 | ⊢ (x = B → (φ ↔ ψ)) |
Ref | Expression |
---|---|
eqreu | ⊢ ((B ∈ A ∧ ψ ∧ ∀x ∈ A (φ → x = B)) → ∃!x ∈ A φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbiim 2441 | . . . . 5 ⊢ (∀x ∈ A (φ ↔ x = B) ↔ (∀x ∈ A (φ → x = B) ∧ ∀x ∈ A (x = B → φ))) | |
2 | eqreu.1 | . . . . . . 7 ⊢ (x = B → (φ ↔ ψ)) | |
3 | 2 | ceqsralv 2579 | . . . . . 6 ⊢ (B ∈ A → (∀x ∈ A (x = B → φ) ↔ ψ)) |
4 | 3 | anbi2d 437 | . . . . 5 ⊢ (B ∈ A → ((∀x ∈ A (φ → x = B) ∧ ∀x ∈ A (x = B → φ)) ↔ (∀x ∈ A (φ → x = B) ∧ ψ))) |
5 | 1, 4 | syl5bb 181 | . . . 4 ⊢ (B ∈ A → (∀x ∈ A (φ ↔ x = B) ↔ (∀x ∈ A (φ → x = B) ∧ ψ))) |
6 | reu6i 2726 | . . . . 5 ⊢ ((B ∈ A ∧ ∀x ∈ A (φ ↔ x = B)) → ∃!x ∈ A φ) | |
7 | 6 | ex 108 | . . . 4 ⊢ (B ∈ A → (∀x ∈ A (φ ↔ x = B) → ∃!x ∈ A φ)) |
8 | 5, 7 | sylbird 159 | . . 3 ⊢ (B ∈ A → ((∀x ∈ A (φ → x = B) ∧ ψ) → ∃!x ∈ A φ)) |
9 | 8 | 3impib 1101 | . 2 ⊢ ((B ∈ A ∧ ∀x ∈ A (φ → x = B) ∧ ψ) → ∃!x ∈ A φ) |
10 | 9 | 3com23 1109 | 1 ⊢ ((B ∈ A ∧ ψ ∧ ∀x ∈ A (φ → x = B)) → ∃!x ∈ A φ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 ∀wral 2300 ∃!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 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-v 2553 |
This theorem is referenced by: (None) |
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