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Theorem 2reu2rex 3438
Description: Double restricted existential uniqueness, analogous to 2eu2ex 2727. (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 3437 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
2 reurex 3437 . . 3 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32reximi 3248 . 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 3144  ∃!wreu 3145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-eu 2652  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151
This theorem is referenced by:  2reu1  3885  2sqreultblem  25957  2sqreunnltblem  25960
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