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Theorem rmo4 3540
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 3058 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 an4 900 . . . . . . . . 9 (((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝜑𝜓)))
3 ancom 465 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑥𝐴))
43anbi1i 733 . . . . . . . . 9 (((𝑥𝐴𝑦𝐴) ∧ (𝜑𝜓)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)))
52, 4bitri 264 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)))
65imbi1i 338 . . . . . . 7 ((((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)) → 𝑥 = 𝑦))
7 impexp 461 . . . . . . 7 ((((𝑦𝐴𝑥𝐴) ∧ (𝜑𝜓)) → 𝑥 = 𝑦) ↔ ((𝑦𝐴𝑥𝐴) → ((𝜑𝜓) → 𝑥 = 𝑦)))
8 impexp 461 . . . . . . 7 (((𝑦𝐴𝑥𝐴) → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
96, 7, 83bitri 286 . . . . . 6 ((((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
109albii 1896 . . . . 5 (∀𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
11 df-ral 3055 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦𝐴 → (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦))))
12 r19.21v 3098 . . . . 5 (∀𝑦𝐴 (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
1310, 11, 123bitr2i 288 . . . 4 (∀𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
1413albii 1896 . . 3 (∀𝑥𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
15 eleq1w 2822 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
16 rmo4.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16anbi12d 749 . . . 4 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
1817mo4 2655 . . 3 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝑦(((𝑥𝐴𝜑) ∧ (𝑦𝐴𝜓)) → 𝑥 = 𝑦))
19 df-ral 3055 . . 3 (∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦)))
2014, 18, 193bitr4i 292 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
211, 20bitri 264 1 (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wcel 2139  ∃*wmo 2608  wral 3050  ∃*wrmo 3053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clel 2756  df-ral 3055  df-rmo 3058
This theorem is referenced by:  reu4  3541  disjor  4786  somo  5221  supmo  8523  infmo  8566  sqrmo  14191  catideu  16537  poslubmo  17347  posglbmo  17348  mgmidmo  17460  lspextmo  19258  evlseu  19718  ply1divmo  24094  tghilberti2  25732  foot  25813  mideu  25829  2sqmo  29958  cvmliftmo  31573  hilbert1.2  32568  poimirlem1  33723  poimirlem13  33735  poimirlem14  33736  poimirlem18  33740  poimirlem21  33743  inecmo  34443  idomsubgmo  38278
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