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

Theorem reusngf 4575
Description: Restricted existential uniqueness over a singleton. (Contributed by AV, 3-Apr-2023.)
Hypotheses
Ref Expression
rexsngf.1 𝑥𝜓
rexsngf.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
reusngf (𝐴𝑉 → (∃!𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem reusngf
Dummy variables 𝑤 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfsbc1v 3743 . . 3 𝑥[𝑐 / 𝑥]𝜑
2 nfsbc1v 3743 . . 3 𝑥[𝑤 / 𝑥]𝜑
3 sbceq1a 3734 . . 3 (𝑥 = 𝑤 → (𝜑[𝑤 / 𝑥]𝜑))
4 dfsbcq 3725 . . 3 (𝑤 = 𝑐 → ([𝑤 / 𝑥]𝜑[𝑐 / 𝑥]𝜑))
51, 2, 3, 4reu8nf 3809 . 2 (∃!𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐)))
6 rexsngf.1 . . . . 5 𝑥𝜓
7 nfcv 2958 . . . . . 6 𝑥{𝐴}
8 nfv 1915 . . . . . . 7 𝑥 𝐴 = 𝑐
91, 8nfim 1897 . . . . . 6 𝑥([𝑐 / 𝑥]𝜑𝐴 = 𝑐)
107, 9nfralw 3192 . . . . 5 𝑥𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)
116, 10nfan 1900 . . . 4 𝑥(𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐))
12 rexsngf.2 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
13 eqeq1 2805 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝑐𝐴 = 𝑐))
1413imbi2d 344 . . . . . 6 (𝑥 = 𝐴 → (([𝑐 / 𝑥]𝜑𝑥 = 𝑐) ↔ ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)))
1514ralbidv 3165 . . . . 5 (𝑥 = 𝐴 → (∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)))
1612, 15anbi12d 633 . . . 4 (𝑥 = 𝐴 → ((𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐))))
1711, 16rexsngf 4573 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐))))
18 nfv 1915 . . . . . 6 𝑐([𝐴 / 𝑥]𝜑𝐴 = 𝐴)
19 dfsbcq 3725 . . . . . . 7 (𝑐 = 𝐴 → ([𝑐 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
20 eqeq2 2813 . . . . . . 7 (𝑐 = 𝐴 → (𝐴 = 𝑐𝐴 = 𝐴))
2119, 20imbi12d 348 . . . . . 6 (𝑐 = 𝐴 → (([𝑐 / 𝑥]𝜑𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑𝐴 = 𝐴)))
2218, 21ralsngf 4574 . . . . 5 (𝐴𝑉 → (∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑𝐴 = 𝐴)))
2322anbi2d 631 . . . 4 (𝐴𝑉 → ((𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)) ↔ (𝜓 ∧ ([𝐴 / 𝑥]𝜑𝐴 = 𝐴))))
24 eqidd 2802 . . . . 5 ([𝐴 / 𝑥]𝜑𝐴 = 𝐴)
2524biantru 533 . . . 4 (𝜓 ↔ (𝜓 ∧ ([𝐴 / 𝑥]𝜑𝐴 = 𝐴)))
2623, 25syl6bbr 292 . . 3 (𝐴𝑉 → ((𝜓 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝐴 = 𝑐)) ↔ 𝜓))
2717, 26bitrd 282 . 2 (𝐴𝑉 → (∃𝑥 ∈ {𝐴} (𝜑 ∧ ∀𝑐 ∈ {𝐴} ([𝑐 / 𝑥]𝜑𝑥 = 𝑐)) ↔ 𝜓))
285, 27syl5bb 286 1 (𝐴𝑉 → (∃!𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2112  wral 3109  wrex 3110  ∃!wreu 3111  [wsbc 3723  {csn 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-reu 3116  df-v 3446  df-sbc 3724  df-sn 4529
This theorem is referenced by:  reusng  4578  rmosn  4618
  Copyright terms: Public domain W3C validator