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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. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
reuxfr.1 | ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) |
reuxfr.2 | ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Ref | Expression |
---|---|
reuxfr | ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuxfr.1 | . . . 4 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
3 | reuxfr.2 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) |
5 | 2, 4 | reuxfrd 3739 | . 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑)) |
6 | 5 | mptru 1540 | 1 ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∃wrex 3064 ∃!wreu 3368 ∃*wrmo 3369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 |
This theorem is referenced by: (None) |
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