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Theorem rmo2ilem 2970
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 1431 . . . 4 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
3 df-ral 2398 . . . 4 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
42, 3bitr4i 186 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
54exbii 1569 . 2 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
6 nfv 1493 . . . . 5 𝑦 𝑥𝐴
7 rmo2.1 . . . . 5 𝑦𝜑
86, 7nfan 1529 . . . 4 𝑦(𝑥𝐴𝜑)
98mo2r 2029 . . 3 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) → ∃*𝑥(𝑥𝐴𝜑))
10 df-rmo 2401 . . 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 1314   = wceq 1316  wnf 1421  wex 1453  wcel 1465  ∃*wmo 1978  wral 2393  ∃*wrmo 2396
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-ral 2398  df-rmo 2401
This theorem is referenced by:  rmo2i  2971
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