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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 30240 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | |
2 | 2rexreu 3749 | . 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃wrex 3138 ∃!wreu 3139 ∃!w2reu 30239 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-mo 2621 df-eu 2653 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-2reu 30240 |
This theorem is referenced by: (None) |
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