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Theorem 2reu2rex 40487
Description: Double restricted existential uniqueness, analogous to 2eu2ex 2545. (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 3149 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
2 reurex 3149 . . 3 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32reximi 3005 . 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 2908  ∃!wreu 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-eu 2473  df-mo 2474  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915
This theorem is referenced by:  2reu1  40490
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