MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmo2 Structured version   Visualization version   GIF version

Theorem rmo2 3896
Description: Alternate definition of restricted "at most one". Note that ∃*𝑥𝐴𝜑 is not equivalent to 𝑦𝐴𝑥𝐴(𝜑𝑥 = 𝑦) (in analogy to reu6 3735); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3897. (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 3378 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 nfv 1912 . . . 4 𝑦 𝑥𝐴
3 rmo2.1 . . . 4 𝑦𝜑
42, 3nfan 1897 . . 3 𝑦(𝑥𝐴𝜑)
54mof 2561 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
6 impexp 450 . . . . 5 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
76albii 1816 . . . 4 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
8 df-ral 3060 . . . 4 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
97, 8bitr4i 278 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
109exbii 1845 . 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 206  wa 395  wal 1535  wex 1776  wnf 1780  wcel 2106  ∃*wmo 2536  wral 3059  ∃*wrmo 3377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-ral 3060  df-rmo 3378
This theorem is referenced by:  rmo2i  3897  rmoanimALT  3904  disjiun  5136  poimirlem2  37609  onsucf1lem  43259
  Copyright terms: Public domain W3C validator