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Theorem rmo2 3874
Description: Alternate definition of restricted "at most one." Note that ∃*𝑥𝐴𝜑 is not equivalent to 𝑦𝐴𝑥𝐴(𝜑𝑥 = 𝑦) (in analogy to reu6 3721); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3875. (Contributed by NM, 17-Jun-2017.)
Hypothesis
Ref Expression
rmo2.1 𝑦𝜑
Assertion
Ref Expression
rmo2 (∃*𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rmo2
StepHypRef Expression
1 df-rmo 3151 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 nfv 1908 . . . 4 𝑦 𝑥𝐴
3 rmo2.1 . . . 4 𝑦𝜑
42, 3nfan 1893 . . 3 𝑦(𝑥𝐴𝜑)
54mof 2645 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
6 impexp 451 . . . . 5 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
76albii 1813 . . . 4 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
8 df-ral 3148 . . . 4 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
97, 8bitr4i 279 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
109exbii 1841 . 2 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
111, 5, 103bitri 298 1 (∃*𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528  wex 1773  wnf 1777  wcel 2107  ∃*wmo 2618  wral 3143  ∃*wrmo 3146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-mo 2620  df-ral 3148  df-rmo 3151
This theorem is referenced by:  rmo2i  3875  rmoanimALT  3883  disjiun  5050  poimirlem2  34761
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