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| Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5419 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 481 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) | 
| 3 | reuxfr1.2 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
| 4 | 3 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | 
| 5 | reuxfr1.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 6 | 2, 4, 5 | reuxfr1ds 3756 | . 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓)) | 
| 7 | 6 | mptru 1546 | 1 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 ∃!wreu 3377 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 | 
| This theorem is referenced by: zmax 12988 rebtwnz 12990 | 
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