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Theorem rmo2ilem 2950
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 1414 . . . 4 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
3 df-ral 2380 . . . 4 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
42, 3bitr4i 186 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
54exbii 1552 . 2 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
6 nfv 1476 . . . . 5 𝑦 𝑥𝐴
7 rmo2.1 . . . . 5 𝑦𝜑
86, 7nfan 1512 . . . 4 𝑦(𝑥𝐴𝜑)
98mo2r 2012 . . 3 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) → ∃*𝑥(𝑥𝐴𝜑))
10 df-rmo 2383 . . 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 1297   = wceq 1299  wnf 1404  wex 1436  wcel 1448  ∃*wmo 1961  wral 2375  ∃*wrmo 2378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-ral 2380  df-rmo 2383
This theorem is referenced by:  rmo2i  2951
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