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| Mirrors > Home > MPE Home > Th. List > reubiia | Structured version Visualization version GIF version | ||
| Description: Formula-building rule for restricted existential uniqueness quantifier (inference form). (Contributed by NM, 14-Nov-2004.) | 
| Ref | Expression | 
|---|---|
| rmobiia.1 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| reubiia | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rmobiia.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | pm5.32i 574 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) | 
| 3 | 2 | eubii 2585 | . 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | 
| 4 | df-reu 3381 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 5 | df-reu 3381 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
| 6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∃!weu 2568 ∃!wreu 3378 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-mo 2540 df-eu 2569 df-reu 3381 | 
| This theorem is referenced by: reubii 3389 reuanid 3391 riotaxfrd 7422 opreuopreu 8059 infempty 9547 | 
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