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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 2reu7 | Structured version Visualization version GIF version | ||
| Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2658. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
| Ref | Expression |
|---|---|
| 2reu7 | ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 2 | nfre1 3285 | . . . 4 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑 | |
| 3 | 1, 2 | nfreuw 3414 | . . 3 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 |
| 4 | 3 | reuan 3896 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)) |
| 5 | ancom 460 | . . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑)) | |
| 6 | 5 | reubii 3389 | . . . 4 ⊢ (∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑦 ∈ 𝐵 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑)) |
| 7 | nfre1 3285 | . . . . 5 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑 | |
| 8 | 7 | reuan 3896 | . . . 4 ⊢ (∃!𝑦 ∈ 𝐵 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) |
| 9 | ancom 460 | . . . 4 ⊢ ((∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑)) | |
| 10 | 6, 8, 9 | 3bitri 297 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑)) |
| 11 | 10 | reubii 3389 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑)) |
| 12 | ancom 460 | . 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)) | |
| 13 | 4, 11, 12 | 3bitr4ri 304 | 1 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wrex 3070 ∃!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 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 |
| This theorem is referenced by: 2reu8 47124 |
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