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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 3267 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑)) |
3 | 2 | imp 444 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥𝜑) |
4 | 3 | 3adant3 1074 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃𝑥𝜑) |
5 | eqeq2 2621 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
6 | 5 | imbi2d 329 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴))) |
7 | 6 | albidv 1836 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴))) |
8 | 7 | spcegv 3267 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
9 | 8 | imp 444 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
10 | 9 | 3adant2 1073 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
11 | eu3v 2486 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | |
12 | 4, 10, 11 | sylanbrc 695 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃!weu 2458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 |
This theorem is referenced by: rngurd 29119 neibastop3 31527 upixp 32694 zrinitorngc 41792 zrtermorngc 41793 zrtermoringc 41862 |
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