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Theorem zfac 9876
Description: Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 9875. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfac 𝑥𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfac
Dummy variables 𝑣 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-ac 9875 . 2 𝑥𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑣𝑢(∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑣))
2 equequ2 2026 . . . . . . . . . 10 (𝑣 = 𝑤 → (𝑢 = 𝑣𝑢 = 𝑤))
32bibi2d 344 . . . . . . . . 9 (𝑣 = 𝑤 → ((∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑣) ↔ (∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑤)))
4 elequ2 2122 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (𝑧𝑡𝑧𝑤))
54anbi2d 628 . . . . . . . . . . . 12 (𝑡 = 𝑤 → ((𝑢𝑧𝑧𝑡) ↔ (𝑢𝑧𝑧𝑤)))
6 elequ2 2122 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
7 elequ1 2114 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (𝑡𝑥𝑤𝑥))
86, 7anbi12d 630 . . . . . . . . . . . 12 (𝑡 = 𝑤 → ((𝑢𝑡𝑡𝑥) ↔ (𝑢𝑤𝑤𝑥)))
95, 8anbi12d 630 . . . . . . . . . . 11 (𝑡 = 𝑤 → (((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ ((𝑢𝑧𝑧𝑤) ∧ (𝑢𝑤𝑤𝑥))))
109cbvexvw 2037 . . . . . . . . . 10 (∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ ∃𝑤((𝑢𝑧𝑧𝑤) ∧ (𝑢𝑤𝑤𝑥)))
1110bibi1i 340 . . . . . . . . 9 ((∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑤) ↔ (∃𝑤((𝑢𝑧𝑧𝑤) ∧ (𝑢𝑤𝑤𝑥)) ↔ 𝑢 = 𝑤))
123, 11syl6bb 288 . . . . . . . 8 (𝑣 = 𝑤 → ((∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑣) ↔ (∃𝑤((𝑢𝑧𝑧𝑤) ∧ (𝑢𝑤𝑤𝑥)) ↔ 𝑢 = 𝑤)))
1312albidv 1914 . . . . . . 7 (𝑣 = 𝑤 → (∀𝑢(∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑣) ↔ ∀𝑢(∃𝑤((𝑢𝑧𝑧𝑤) ∧ (𝑢𝑤𝑤𝑥)) ↔ 𝑢 = 𝑤)))
14 elequ1 2114 . . . . . . . . . . . 12 (𝑢 = 𝑦 → (𝑢𝑧𝑦𝑧))
1514anbi1d 629 . . . . . . . . . . 11 (𝑢 = 𝑦 → ((𝑢𝑧𝑧𝑤) ↔ (𝑦𝑧𝑧𝑤)))
16 elequ1 2114 . . . . . . . . . . . 12 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
1716anbi1d 629 . . . . . . . . . . 11 (𝑢 = 𝑦 → ((𝑢𝑤𝑤𝑥) ↔ (𝑦𝑤𝑤𝑥)))
1815, 17anbi12d 630 . . . . . . . . . 10 (𝑢 = 𝑦 → (((𝑢𝑧𝑧𝑤) ∧ (𝑢𝑤𝑤𝑥)) ↔ ((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥))))
1918exbidv 1915 . . . . . . . . 9 (𝑢 = 𝑦 → (∃𝑤((𝑢𝑧𝑧𝑤) ∧ (𝑢𝑤𝑤𝑥)) ↔ ∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥))))
20 equequ1 2025 . . . . . . . . 9 (𝑢 = 𝑦 → (𝑢 = 𝑤𝑦 = 𝑤))
2119, 20bibi12d 347 . . . . . . . 8 (𝑢 = 𝑦 → ((∃𝑤((𝑢𝑧𝑧𝑤) ∧ (𝑢𝑤𝑤𝑥)) ↔ 𝑢 = 𝑤) ↔ (∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
2221cbvalvw 2036 . . . . . . 7 (∀𝑢(∃𝑤((𝑢𝑧𝑧𝑤) ∧ (𝑢𝑤𝑤𝑥)) ↔ 𝑢 = 𝑤) ↔ ∀𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
2313, 22syl6bb 288 . . . . . 6 (𝑣 = 𝑤 → (∀𝑢(∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑣) ↔ ∀𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
2423cbvexvw 2037 . . . . 5 (∃𝑣𝑢(∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑣) ↔ ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
2524imbi2i 337 . . . 4 (((𝑦𝑧𝑧𝑤) → ∃𝑣𝑢(∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑣)) ↔ ((𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
26252albii 1814 . . 3 (∀𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑣𝑢(∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑣)) ↔ ∀𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
2726exbii 1841 . 2 (∃𝑥𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑣𝑢(∃𝑡((𝑢𝑧𝑧𝑡) ∧ (𝑢𝑡𝑡𝑥)) ↔ 𝑢 = 𝑣)) ↔ ∃𝑥𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
281, 27mpbi 231 1 𝑥𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ac 9875
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774
This theorem is referenced by:  axacndlem4  10026
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