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Theorem rmo2i 3493
 Description: Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
Hypothesis
Ref Expression
rmo2.1 𝑦𝜑
Assertion
Ref Expression
rmo2i (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rmo2i
StepHypRef Expression
1 rexex 2985 . 2 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
2 rmo2.1 . . 3 𝑦𝜑
32rmo2 3492 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
41, 3sylibr 223 1 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1695  Ⅎwnf 1699  ∀wral 2896  ∃wrex 2897  ∃*wrmo 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-ral 2901  df-rex 2902  df-rmo 2904 This theorem is referenced by: (None)
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