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

Theorem euind 3670
Description: Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
Hypotheses
Ref Expression
euind.1 𝐵 ∈ V
euind.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
euind ((∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧𝑥(𝜑𝑧 = 𝐴))
Distinct variable groups:   𝑦,𝑧,𝜑   𝑥,𝑧,𝜓   𝑦,𝐴,𝑧   𝑥,𝐵,𝑧   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem euind
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 euind.2 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
21cbvexvw 2039 . . . . 5 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
3 euind.1 . . . . . . . . 9 𝐵 ∈ V
43isseti 3456 . . . . . . . 8 𝑧 𝑧 = 𝐵
54biantrur 531 . . . . . . 7 (𝜓 ↔ (∃𝑧 𝑧 = 𝐵𝜓))
65exbii 1849 . . . . . 6 (∃𝑦𝜓 ↔ ∃𝑦(∃𝑧 𝑧 = 𝐵𝜓))
7 19.41v 1952 . . . . . . 7 (∃𝑧(𝑧 = 𝐵𝜓) ↔ (∃𝑧 𝑧 = 𝐵𝜓))
87exbii 1849 . . . . . 6 (∃𝑦𝑧(𝑧 = 𝐵𝜓) ↔ ∃𝑦(∃𝑧 𝑧 = 𝐵𝜓))
9 excom 2161 . . . . . 6 (∃𝑦𝑧(𝑧 = 𝐵𝜓) ↔ ∃𝑧𝑦(𝑧 = 𝐵𝜓))
106, 8, 93bitr2i 298 . . . . 5 (∃𝑦𝜓 ↔ ∃𝑧𝑦(𝑧 = 𝐵𝜓))
112, 10bitri 274 . . . 4 (∃𝑥𝜑 ↔ ∃𝑧𝑦(𝑧 = 𝐵𝜓))
12 eqeq2 2748 . . . . . . . . 9 (𝐴 = 𝐵 → (𝑧 = 𝐴𝑧 = 𝐵))
1312imim2i 16 . . . . . . . 8 (((𝜑𝜓) → 𝐴 = 𝐵) → ((𝜑𝜓) → (𝑧 = 𝐴𝑧 = 𝐵)))
14 biimpr 219 . . . . . . . . . 10 ((𝑧 = 𝐴𝑧 = 𝐵) → (𝑧 = 𝐵𝑧 = 𝐴))
1514imim2i 16 . . . . . . . . 9 (((𝜑𝜓) → (𝑧 = 𝐴𝑧 = 𝐵)) → ((𝜑𝜓) → (𝑧 = 𝐵𝑧 = 𝐴)))
16 an31 645 . . . . . . . . . . 11 (((𝜑𝜓) ∧ 𝑧 = 𝐵) ↔ ((𝑧 = 𝐵𝜓) ∧ 𝜑))
1716imbi1i 349 . . . . . . . . . 10 ((((𝜑𝜓) ∧ 𝑧 = 𝐵) → 𝑧 = 𝐴) ↔ (((𝑧 = 𝐵𝜓) ∧ 𝜑) → 𝑧 = 𝐴))
18 impexp 451 . . . . . . . . . 10 ((((𝜑𝜓) ∧ 𝑧 = 𝐵) → 𝑧 = 𝐴) ↔ ((𝜑𝜓) → (𝑧 = 𝐵𝑧 = 𝐴)))
19 impexp 451 . . . . . . . . . 10 ((((𝑧 = 𝐵𝜓) ∧ 𝜑) → 𝑧 = 𝐴) ↔ ((𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)))
2017, 18, 193bitr3i 300 . . . . . . . . 9 (((𝜑𝜓) → (𝑧 = 𝐵𝑧 = 𝐴)) ↔ ((𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)))
2115, 20sylib 217 . . . . . . . 8 (((𝜑𝜓) → (𝑧 = 𝐴𝑧 = 𝐵)) → ((𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)))
2213, 21syl 17 . . . . . . 7 (((𝜑𝜓) → 𝐴 = 𝐵) → ((𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)))
23222alimi 1813 . . . . . 6 (∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) → ∀𝑥𝑦((𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)))
24 19.23v 1944 . . . . . . . 8 (∀𝑦((𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)) ↔ (∃𝑦(𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)))
2524albii 1820 . . . . . . 7 (∀𝑥𝑦((𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)) ↔ ∀𝑥(∃𝑦(𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)))
26 19.21v 1941 . . . . . . 7 (∀𝑥(∃𝑦(𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)) ↔ (∃𝑦(𝑧 = 𝐵𝜓) → ∀𝑥(𝜑𝑧 = 𝐴)))
2725, 26bitri 274 . . . . . 6 (∀𝑥𝑦((𝑧 = 𝐵𝜓) → (𝜑𝑧 = 𝐴)) ↔ (∃𝑦(𝑧 = 𝐵𝜓) → ∀𝑥(𝜑𝑧 = 𝐴)))
2823, 27sylib 217 . . . . 5 (∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) → (∃𝑦(𝑧 = 𝐵𝜓) → ∀𝑥(𝜑𝑧 = 𝐴)))
2928eximdv 1919 . . . 4 (∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) → (∃𝑧𝑦(𝑧 = 𝐵𝜓) → ∃𝑧𝑥(𝜑𝑧 = 𝐴)))
3011, 29biimtrid 241 . . 3 (∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) → (∃𝑥𝜑 → ∃𝑧𝑥(𝜑𝑧 = 𝐴)))
3130imp 407 . 2 ((∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃𝑧𝑥(𝜑𝑧 = 𝐴))
32 pm4.24 564 . . . . . . . . 9 (𝜑 ↔ (𝜑𝜑))
3332biimpi 215 . . . . . . . 8 (𝜑 → (𝜑𝜑))
34 anim12 806 . . . . . . . 8 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝑤 = 𝐴)) → ((𝜑𝜑) → (𝑧 = 𝐴𝑤 = 𝐴)))
35 eqtr3 2762 . . . . . . . 8 ((𝑧 = 𝐴𝑤 = 𝐴) → 𝑧 = 𝑤)
3633, 34, 35syl56 36 . . . . . . 7 (((𝜑𝑧 = 𝐴) ∧ (𝜑𝑤 = 𝐴)) → (𝜑𝑧 = 𝑤))
3736alanimi 1817 . . . . . 6 ((∀𝑥(𝜑𝑧 = 𝐴) ∧ ∀𝑥(𝜑𝑤 = 𝐴)) → ∀𝑥(𝜑𝑧 = 𝑤))
38 19.23v 1944 . . . . . 6 (∀𝑥(𝜑𝑧 = 𝑤) ↔ (∃𝑥𝜑𝑧 = 𝑤))
3937, 38sylib 217 . . . . 5 ((∀𝑥(𝜑𝑧 = 𝐴) ∧ ∀𝑥(𝜑𝑤 = 𝐴)) → (∃𝑥𝜑𝑧 = 𝑤))
4039com12 32 . . . 4 (∃𝑥𝜑 → ((∀𝑥(𝜑𝑧 = 𝐴) ∧ ∀𝑥(𝜑𝑤 = 𝐴)) → 𝑧 = 𝑤))
4140alrimivv 1930 . . 3 (∃𝑥𝜑 → ∀𝑧𝑤((∀𝑥(𝜑𝑧 = 𝐴) ∧ ∀𝑥(𝜑𝑤 = 𝐴)) → 𝑧 = 𝑤))
4241adantl 482 . 2 ((∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∀𝑧𝑤((∀𝑥(𝜑𝑧 = 𝐴) ∧ ∀𝑥(𝜑𝑤 = 𝐴)) → 𝑧 = 𝑤))
43 eqeq1 2740 . . . . 5 (𝑧 = 𝑤 → (𝑧 = 𝐴𝑤 = 𝐴))
4443imbi2d 340 . . . 4 (𝑧 = 𝑤 → ((𝜑𝑧 = 𝐴) ↔ (𝜑𝑤 = 𝐴)))
4544albidv 1922 . . 3 (𝑧 = 𝑤 → (∀𝑥(𝜑𝑧 = 𝐴) ↔ ∀𝑥(𝜑𝑤 = 𝐴)))
4645eu4 2615 . 2 (∃!𝑧𝑥(𝜑𝑧 = 𝐴) ↔ (∃𝑧𝑥(𝜑𝑧 = 𝐴) ∧ ∀𝑧𝑤((∀𝑥(𝜑𝑧 = 𝐴) ∧ ∀𝑥(𝜑𝑤 = 𝐴)) → 𝑧 = 𝑤)))
4731, 42, 46sylanbrc 583 1 ((∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧𝑥(𝜑𝑧 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1538   = wceq 1540  wex 1780  wcel 2105  ∃!weu 2566  Vcvv 3441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-mo 2538  df-eu 2567  df-cleq 2728  df-clel 2814
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator