MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmo4 Structured version   Visualization version   GIF version

Theorem rmo4 3723
Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
rmo4.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
rmo4 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rmo4
StepHypRef Expression
1 df-rmo 3371 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 an4 655 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑𝜓)))
3 ancom 460 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑥𝐴))
43anbi1i 623 . . . . . . . . 9 (((𝑥𝐴𝑦𝐴) ∧ (𝜑𝜓)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)))
52, 4bitri 275 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)))
65imbi1i 349 . . . . . . 7 ((((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)) → 𝑥 = 𝑦))
7 impexp 450 . . . . . . 7 ((((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)) → 𝑥 = 𝑦) ↔ ((𝑦𝐴𝑥𝐴) → ((𝜑𝜓) → 𝑥 = 𝑦)))
8 impexp 450 . . . . . . 7 (((𝑦𝐴𝑥𝐴) → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
96, 7, 83bitri 297 . . . . . 6 ((((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
109albii 1814 . . . . 5 (∀𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
11 df-ral 3057 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
12 r19.21v 3174 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
1310, 11, 123bitr2i 299 . . . 4 (∀𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
1413albii 1814 . . 3 (∀𝑥𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
15 eleq1w 2811 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
16 rmo4.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16anbi12d 630 . . . 4 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
1817mo4 2555 . . 3 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦))
19 df-ral 3057 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
2014, 18, 193bitr4i 303 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
211, 20bitri 275 1 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1532  wcel 2099  ∃*wmo 2527  wral 3056  ∃*wrmo 3370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-mo 2529  df-clel 2805  df-ral 3057  df-rmo 3371
This theorem is referenced by:  reu4  3724  disjor  5122  somo  5621  nnasmo  8677  supmo  9467  infmo  9510  sqrmo  15222  catideu  17646  poslubmo  18394  posglbmo  18395  mgmidmo  18611  mndinvmod  18715  lspextmo  20930  evlseu  22016  ply1divmo  26058  2sqmo  27357  divsmo  28071  tghilberti2  28429  foot  28513  mideu  28529  cvmliftmo  34830  hilbert1.2  35687  poimirlem1  37029  poimirlem13  37041  poimirlem14  37042  poimirlem18  37046  poimirlem21  37049  inecmo  37763  idomsubgmo  42543
  Copyright terms: Public domain W3C validator