MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  aceq2 Structured version   Visualization version   GIF version

Theorem aceq2 9055
Description: Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
aceq2 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢

Proof of Theorem aceq2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 df-ral 3019 . . . . 5 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2 19.23v 1984 . . . . 5 (∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
31, 2bitri 264 . . . 4 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
4 biidd 252 . . . . 5 (𝑤 = 𝑡 → (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
54cbvralv 3274 . . . 4 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
6 n0 4039 . . . . 5 (𝑧 ≠ ∅ ↔ ∃𝑡 𝑡𝑧)
7 elequ2 2117 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑧𝑣𝑧𝑢))
8 elequ2 2117 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑤𝑣𝑤𝑢))
97, 8anbi12d 749 . . . . . . . 8 (𝑣 = 𝑢 → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑢𝑤𝑢)))
109cbvrexv 3275 . . . . . . 7 (∃𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃𝑢𝑦 (𝑧𝑢𝑤𝑢))
1110reubii 3231 . . . . . 6 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢))
12 eleq1 2791 . . . . . . . . 9 (𝑤 = 𝑣 → (𝑤𝑢𝑣𝑢))
1312anbi2d 742 . . . . . . . 8 (𝑤 = 𝑣 → ((𝑧𝑢𝑤𝑢) ↔ (𝑧𝑢𝑣𝑢)))
1413rexbidv 3154 . . . . . . 7 (𝑤 = 𝑣 → (∃𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
1514cbvreuv 3276 . . . . . 6 (∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
1611, 15bitri 264 . . . . 5 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
176, 16imbi12i 339 . . . 4 ((𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
183, 5, 173bitr4i 292 . . 3 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
1918ralbii 3082 . 2 (∀𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
2019exbii 1887 1 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1594  wex 1817  wne 2896  wral 3014  wrex 3015  ∃!wreu 3016  c0 4023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-v 3306  df-dif 3683  df-nul 4024
This theorem is referenced by:  dfac7  9067  ac3  9397
  Copyright terms: Public domain W3C validator