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Theorem aceq2 9163
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 3069 . . . . 5 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2 19.23v 2026 . . . . 5 (∀𝑡(𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
31, 2bitri 265 . . . 4 (∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
4 biidd 253 . . . . 5 (𝑤 = 𝑡 → (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
54cbvralv 3324 . . . 4 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑡𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
6 n0 4089 . . . . 5 (𝑧 ≠ ∅ ↔ ∃𝑡 𝑡𝑧)
7 elequ2 2162 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑧𝑣𝑧𝑢))
8 elequ2 2162 . . . . . . . . 9 (𝑣 = 𝑢 → (𝑤𝑣𝑤𝑢))
97, 8anbi12d 617 . . . . . . . 8 (𝑣 = 𝑢 → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑢𝑤𝑢)))
109cbvrexv 3325 . . . . . . 7 (∃𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃𝑢𝑦 (𝑧𝑢𝑤𝑢))
1110reubii 3281 . . . . . 6 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢))
12 eleq1w 2836 . . . . . . . . 9 (𝑤 = 𝑣 → (𝑤𝑢𝑣𝑢))
1312anbi2d 615 . . . . . . . 8 (𝑤 = 𝑣 → ((𝑧𝑢𝑤𝑢) ↔ (𝑧𝑢𝑣𝑢)))
1413rexbidv 3204 . . . . . . 7 (𝑤 = 𝑣 → (∃𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
1514cbvreuv 3326 . . . . . 6 (∃!𝑤𝑧𝑢𝑦 (𝑧𝑢𝑤𝑢) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
1611, 15bitri 265 . . . . 5 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
176, 16imbi12i 340 . . . 4 ((𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ↔ (∃𝑡 𝑡𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
183, 5, 173bitr4i 293 . . 3 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
1918ralbii 3132 . 2 (∀𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
2019exbii 1927 1 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383  wal 1632  wex 1855  wne 2946  wral 3064  wrex 3065  ∃!wreu 3066  c0 4073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-v 3357  df-dif 3732  df-nul 4074
This theorem is referenced by:  dfac7  9177  ac3  9507
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