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Theorem 2reu2rex 3342
 Description: Double restricted existential uniqueness, analogous to 2eu2ex 2664. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
2reu2rex (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem 2reu2rex
StepHypRef Expression
1 reurex 3341 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
2 reurex 3341 . . 3 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32reximi 3171 . 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 3071  ∃!wreu 3072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-eu 2588  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078 This theorem is referenced by:  2reu1  3805  2sqreultblem  26144  2sqreunnltblem  26147
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