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Theorem rmo2ilem 3044
Description: Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.)
Hypothesis
Ref Expression
rmo2.1 𝑦𝜑
Assertion
Ref Expression
rmo2ilem (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rmo2ilem
StepHypRef Expression
1 impexp 261 . . . . 5 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
21albii 1463 . . . 4 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
3 df-ral 2453 . . . 4 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
42, 3bitr4i 186 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
54exbii 1598 . 2 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
6 nfv 1521 . . . . 5 𝑦 𝑥𝐴
7 rmo2.1 . . . . 5 𝑦𝜑
86, 7nfan 1558 . . . 4 𝑦(𝑥𝐴𝜑)
98mo2r 2071 . . 3 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) → ∃*𝑥(𝑥𝐴𝜑))
10 df-rmo 2456 . . 3 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
119, 10sylibr 133 . 2 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
125, 11sylbir 134 1 (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1346   = wceq 1348  wnf 1453  wex 1485  ∃*wmo 2020  wcel 2141  wral 2448  ∃*wrmo 2451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-ral 2453  df-rmo 2456
This theorem is referenced by:  rmo2i  3045
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