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Theorem rmo2 3820
Description: Alternate definition of restricted "at most one". Note that ∃*𝑥𝐴𝜑 is not equivalent to 𝑦𝐴𝑥𝐴(𝜑𝑥 = 𝑦) (in analogy to reu6 3661); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3821. (Contributed by NM, 17-Jun-2017.)
Hypothesis
Ref Expression
rmo2.1 𝑦𝜑
Assertion
Ref Expression
rmo2 (∃*𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem rmo2
StepHypRef Expression
1 df-rmo 3071 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 nfv 1917 . . . 4 𝑦 𝑥𝐴
3 rmo2.1 . . . 4 𝑦𝜑
42, 3nfan 1902 . . 3 𝑦(𝑥𝐴𝜑)
54mof 2563 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
6 impexp 451 . . . . 5 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
76albii 1822 . . . 4 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
8 df-ral 3069 . . . 4 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
97, 8bitr4i 277 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
109exbii 1850 . 2 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
111, 5, 103bitri 297 1 (∃*𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1537  wex 1782  wnf 1786  wcel 2106  ∃*wmo 2538  wral 3064  ∃*wrmo 3067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-mo 2540  df-ral 3069  df-rmo 3071
This theorem is referenced by:  rmo2i  3821  rmoanimALT  3828  disjiun  5061  poimirlem2  35779
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