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Theorem rmo2i 3076
Description: Condition implying restricted at-most-one quantifier. (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 2540 . 2 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
2 rmo2.1 . . 3 𝑦𝜑
32rmo2ilem 3075 . 2 (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
41, 3syl 14 1 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wnf 1471  wex 1503  wral 2472  wrex 2473  ∃*wrmo 2475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-ral 2477  df-rex 2478  df-rmo 2480
This theorem is referenced by: (None)
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