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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2reu2reu2 | Structured version Visualization version GIF version |
Description: Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023.) |
Ref | Expression |
---|---|
2reu2reu2 | ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2reu 32507 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | |
2 | 2rexreu 3771 | . 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | |
3 | 1, 2 | sylbi 217 | 1 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wrex 3068 ∃!wreu 3376 ∃!w2reu 32506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-10 2139 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-mo 2538 df-eu 2567 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-2reu 32507 |
This theorem is referenced by: (None) |
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