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Mirrors > Home > MPE Home > Th. List > reuxfr1 | Structured version Visualization version GIF version |
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5418 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.) |
Ref | Expression |
---|---|
reuxfr1.1 | ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) |
reuxfr1.2 | ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
reuxfr1.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
reuxfr1 | ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuxfr1.1 | . . . 4 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) | |
2 | 1 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
3 | reuxfr1.2 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
4 | 3 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
5 | reuxfr1.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 2, 4, 5 | reuxfr1ds 3747 | . 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓)) |
7 | 6 | mptru 1549 | 1 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ∃!wreu 3375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 |
This theorem is referenced by: zmax 12926 rebtwnz 12928 |
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