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Theorem rmo4 3558
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 3063 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 an4 646 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑𝜓)))
3 ancom 452 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑥𝐴))
43anbi1i 617 . . . . . . . . 9 (((𝑥𝐴𝑦𝐴) ∧ (𝜑𝜓)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)))
52, 4bitri 266 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)))
65imbi1i 340 . . . . . . 7 ((((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)) → 𝑥 = 𝑦))
7 impexp 441 . . . . . . 7 ((((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)) → 𝑥 = 𝑦) ↔ ((𝑦𝐴𝑥𝐴) → ((𝜑𝜓) → 𝑥 = 𝑦)))
8 impexp 441 . . . . . . 7 (((𝑦𝐴𝑥𝐴) → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
96, 7, 83bitri 288 . . . . . 6 ((((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
109albii 1914 . . . . 5 (∀𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
11 df-ral 3060 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
12 r19.21v 3107 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
1310, 11, 123bitr2i 290 . . . 4 (∀𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
1413albii 1914 . . 3 (∀𝑥𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
15 eleq1w 2827 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
16 rmo4.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16anbi12d 624 . . . 4 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
1817mo4 2638 . . 3 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦))
19 df-ral 3060 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
2014, 18, 193bitr4i 294 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
211, 20bitri 266 1 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wcel 2155  ∃*wmo 2563  wral 3055  ∃*wrmo 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-clel 2761  df-ral 3060  df-rmo 3063
This theorem is referenced by:  reu4  3559  disjor  4791  somo  5232  supmo  8565  infmo  8608  sqrmo  14279  catideu  16603  poslubmo  17414  posglbmo  17415  mgmidmo  17527  lspextmo  19328  evlseu  19789  ply1divmo  24186  tghilberti2  25824  foot  25905  mideu  25921  2sqmo  30031  cvmliftmo  31646  hilbert1.2  32638  poimirlem1  33766  poimirlem13  33778  poimirlem14  33779  poimirlem18  33783  poimirlem21  33786  inecmo  34481  idomsubgmo  38385
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