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Mirrors > Home > MPE Home > Th. List > eqeu | Structured version Visualization version GIF version |
Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |
Ref | Expression |
---|---|
eqeu.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
eqeu | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeu.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | spcegv 3482 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑)) |
3 | 2 | imp 396 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥𝜑) |
4 | 3 | 3adant3 1163 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃𝑥𝜑) |
5 | eqeq2 2810 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
6 | 5 | imbi2d 332 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴))) |
7 | 6 | albidv 2016 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴))) |
8 | 7 | spcegv 3482 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
9 | 8 | imp 396 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
10 | 9 | 3adant2 1162 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
11 | eu3v 2610 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | |
12 | 4, 10, 11 | sylanbrc 579 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1108 ∀wal 1651 = wceq 1653 ∃wex 1875 ∈ wcel 2157 ∃!weu 2608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-v 3387 |
This theorem is referenced by: rngurd 30304 neibastop3 32869 upixp 34012 zrinitorngc 42799 zrtermorngc 42800 zrtermoringc 42869 |
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