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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmoeqALT | Structured version Visualization version GIF version |
Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) Obsolete version of rmoeq 3372 as of 27-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rmoeqALT | ⊢ (𝐴 ∈ 𝑉 → ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
2 | 1 | rgenw 2908 | . . 3 ⊢ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝐴) |
3 | eqeq2 2621 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
4 | 3 | imbi2d 329 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝐴 → 𝑥 = 𝑦) ↔ (𝑥 = 𝐴 → 𝑥 = 𝐴))) |
5 | 4 | ralbidv 2969 | . . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝐴))) |
6 | 5 | spcegv 3267 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝐴) → ∃𝑦∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝑦))) |
7 | 2, 6 | mpi 20 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝑦)) |
8 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
9 | 8 | rmo2 3492 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃𝑦∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝑦)) |
10 | 7, 9 | sylibr 223 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃*wrmo 2899 |
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-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-ral 2901 df-rmo 2904 df-v 3175 |
This theorem is referenced by: (None) |
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