Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rmo4f Structured version   Visualization version   GIF version

Theorem rmo4f 29616
Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
rmo4f.1 𝑥𝐴
rmo4f.2 𝑦𝐴
rmo4f.3 𝑥𝜓
rmo4f.4 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
rmo4f (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem rmo4f
StepHypRef Expression
1 rmo4f.1 . . 3 𝑥𝐴
2 rmo4f.2 . . 3 𝑦𝐴
3 nfv 1980 . . 3 𝑦𝜑
41, 2, 3rmo3f 29614 . 2 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
5 rmo4f.3 . . . . . 6 𝑥𝜓
6 rmo4f.4 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
75, 6sbie 2533 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
87anbi2i 732 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓))
98imbi1i 338 . . 3 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝑦))
1092ralbii 3107 . 2 (∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
114, 10bitri 264 1 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wnf 1845  [wsb 2034  wnfc 2877  wral 3038  ∃*wrmo 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rmo 3046
This theorem is referenced by:  disjorf  29670  funcnv5mpt  29749
  Copyright terms: Public domain W3C validator