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Theorem 2reu2reu2 30839
Description: Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023.)
Assertion
Ref Expression
2reu2reu2 (∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reu2reu2
StepHypRef Expression
1 df-2reu 30835 . 2 (∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
2 2rexreu 3696 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
31, 2sylbi 216 1 (∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wrex 3065  ∃!wreu 3066  ∃!w2reu 30834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-10 2137  ax-11 2154  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-mo 2540  df-eu 2569  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-2reu 30835
This theorem is referenced by: (None)
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