Metamath Proof Explorer


Theorem zfcndac

Description: Axiom of Choice ax-ac , reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion zfcndac 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) )

Proof

Step Hyp Ref Expression
1 axacnd 𝑦𝑧𝑤 ( ∀ 𝑦 ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) )
2 19.3v ( ∀ 𝑦 ( 𝑧𝑤𝑤𝑥 ) ↔ ( 𝑧𝑤𝑤𝑥 ) )
3 2 imbi1i ( ( ∀ 𝑦 ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) ↔ ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
4 3 2albii ( ∀ 𝑧𝑤 ( ∀ 𝑦 ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) ↔ ∀ 𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
5 4 exbii ( ∃ 𝑦𝑧𝑤 ( ∀ 𝑦 ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) ↔ ∃ 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
6 1 5 mpbi 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) )
7 equequ2 ( 𝑣 = 𝑥 → ( 𝑢 = 𝑣𝑢 = 𝑥 ) )
8 7 bibi2d ( 𝑣 = 𝑥 → ( ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑥 ) ) )
9 elequ2 ( 𝑡 = 𝑥 → ( 𝑤𝑡𝑤𝑥 ) )
10 9 anbi2d ( 𝑡 = 𝑥 → ( ( 𝑢𝑤𝑤𝑡 ) ↔ ( 𝑢𝑤𝑤𝑥 ) ) )
11 elequ2 ( 𝑡 = 𝑥 → ( 𝑢𝑡𝑢𝑥 ) )
12 elequ1 ( 𝑡 = 𝑥 → ( 𝑡𝑦𝑥𝑦 ) )
13 11 12 anbi12d ( 𝑡 = 𝑥 → ( ( 𝑢𝑡𝑡𝑦 ) ↔ ( 𝑢𝑥𝑥𝑦 ) ) )
14 10 13 anbi12d ( 𝑡 = 𝑥 → ( ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ) )
15 14 cbvexvw ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) )
16 15 bibi1i ( ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑥 ) ↔ ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) )
17 8 16 bitrdi ( 𝑣 = 𝑥 → ( ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) ) )
18 17 albidv ( 𝑣 = 𝑥 → ( ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ∀ 𝑢 ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) ) )
19 elequ1 ( 𝑢 = 𝑧 → ( 𝑢𝑤𝑧𝑤 ) )
20 19 anbi1d ( 𝑢 = 𝑧 → ( ( 𝑢𝑤𝑤𝑥 ) ↔ ( 𝑧𝑤𝑤𝑥 ) ) )
21 elequ1 ( 𝑢 = 𝑧 → ( 𝑢𝑥𝑧𝑥 ) )
22 21 anbi1d ( 𝑢 = 𝑧 → ( ( 𝑢𝑥𝑥𝑦 ) ↔ ( 𝑧𝑥𝑥𝑦 ) ) )
23 20 22 anbi12d ( 𝑢 = 𝑧 → ( ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ) )
24 23 exbidv ( 𝑢 = 𝑧 → ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ) )
25 equequ1 ( 𝑢 = 𝑧 → ( 𝑢 = 𝑥𝑧 = 𝑥 ) )
26 24 25 bibi12d ( 𝑢 = 𝑧 → ( ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) ↔ ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
27 26 cbvalvw ( ∀ 𝑢 ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) ↔ ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) )
28 18 27 bitrdi ( 𝑣 = 𝑥 → ( ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
29 28 cbvexvw ( ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) )
30 29 imbi2i ( ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) ↔ ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
31 30 2albii ( ∀ 𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) ↔ ∀ 𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
32 31 exbii ( ∃ 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) ↔ ∃ 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
33 6 32 mpbir 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) )