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Mirrors > Home > MPE Home > Th. List > reuxfr | Structured version Visualization version GIF version |
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 4822 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.) |
Ref | Expression |
---|---|
reuxfr.1 | ⊢ (𝑦 ∈ 𝐵 → 𝐴 ∈ 𝐵) |
reuxfr.2 | ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐵 𝑥 = 𝐴) |
reuxfr.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
reuxfr | ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuxfr.1 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → 𝐴 ∈ 𝐵) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ 𝐵) |
3 | reuxfr.2 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐵 𝑥 = 𝐴) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = 𝐴) |
5 | reuxfr.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 2, 4, 5 | reuxfrd 4819 | . 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓)) |
7 | 6 | trud 1484 | 1 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∃!wreu 2898 |
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-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-v 3175 |
This theorem is referenced by: zmax 11661 rebtwnz 11663 |
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