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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 482 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
3 | reuxfr.2 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
4 | 3 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) |
5 | 2, 4 | reuxfrd 3736 | . 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑)) |
6 | 5 | mptru 1535 | 1 ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ∃wrex 3136 ∃!wreu 3137 ∃*wrmo 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-cleq 2811 df-clel 2890 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 |
This theorem is referenced by: (None) |
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