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Theorem 2reu2rex 42008
 Description: Double restricted existential uniqueness, analogous to 2eu2ex 2726. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
2reu2rex (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reu2rex
StepHypRef Expression
1 reurex 3372 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
2 reurex 3372 . . 3 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32reximi 3219 . 2 (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)
41, 3syl 17 1 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wrex 3118  ∃!wreu 3119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-eu 2640  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125 This theorem is referenced by:  2reu1  42011
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