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

Theorem axacnd 10085
 Description: A version of the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
axacnd 𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Proof of Theorem axacnd
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 axacndlem5 10084 . . . 4 𝑥𝑦𝑣(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
2 nfnae 2445 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑥
3 nfnae 2445 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑦
4 nfnae 2445 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑤
52, 3, 4nf3an 1902 . . . . 5 𝑥(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤)
6 nfnae 2445 . . . . . . 7 𝑦 ¬ ∀𝑧 𝑧 = 𝑥
7 nfnae 2445 . . . . . . 7 𝑦 ¬ ∀𝑧 𝑧 = 𝑦
8 nfnae 2445 . . . . . . 7 𝑦 ¬ ∀𝑧 𝑧 = 𝑤
96, 7, 8nf3an 1902 . . . . . 6 𝑦(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤)
10 nfnae 2445 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
11 nfnae 2445 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
12 nfnae 2445 . . . . . . . 8 𝑧 ¬ ∀𝑧 𝑧 = 𝑤
1310, 11, 12nf3an 1902 . . . . . . 7 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤)
14 nfcvf 2945 . . . . . . . . . . . 12 (¬ ∀𝑧 𝑧 = 𝑦𝑧𝑦)
15143ad2ant2 1131 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑧𝑦)
16 nfcvd 2920 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑧𝑣)
1715, 16nfeld 2930 . . . . . . . . . 10 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑦𝑣)
18 nfcvf 2945 . . . . . . . . . . . 12 (¬ ∀𝑧 𝑧 = 𝑤𝑧𝑤)
19183ad2ant3 1132 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑧𝑤)
2016, 19nfeld 2930 . . . . . . . . . 10 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑣𝑤)
2117, 20nfand 1898 . . . . . . . . 9 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧(𝑦𝑣𝑣𝑤))
225, 21nfald 2336 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧𝑥(𝑦𝑣𝑣𝑤))
23 nfnae 2445 . . . . . . . . . 10 𝑤 ¬ ∀𝑧 𝑧 = 𝑥
24 nfnae 2445 . . . . . . . . . 10 𝑤 ¬ ∀𝑧 𝑧 = 𝑦
25 nfnae 2445 . . . . . . . . . 10 𝑤 ¬ ∀𝑧 𝑧 = 𝑤
2623, 24, 25nf3an 1902 . . . . . . . . 9 𝑤(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤)
2715, 19nfeld 2930 . . . . . . . . . . . . . 14 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑦𝑤)
28 nfcvf 2945 . . . . . . . . . . . . . . . 16 (¬ ∀𝑧 𝑧 = 𝑥𝑧𝑥)
29283ad2ant1 1130 . . . . . . . . . . . . . . 15 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑧𝑥)
3019, 29nfeld 2930 . . . . . . . . . . . . . 14 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑤𝑥)
3127, 30nfand 1898 . . . . . . . . . . . . 13 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧(𝑦𝑤𝑤𝑥))
3221, 31nfand 1898 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)))
3326, 32nfexd 2337 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)))
3415, 19nfeqd 2929 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧 𝑦 = 𝑤)
3533, 34nfbid 1903 . . . . . . . . . 10 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
369, 35nfald 2336 . . . . . . . . 9 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
3726, 36nfexd 2337 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
3822, 37nfimd 1895 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑧(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
39 nfcvd 2920 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑥𝑣)
40 nfcvf2 2946 . . . . . . . . . . . . 13 (¬ ∀𝑧 𝑧 = 𝑥𝑥𝑧)
41403ad2ant1 1130 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑥𝑧)
4239, 41nfeqd 2929 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑥 𝑣 = 𝑧)
435, 42nfan1 2198 . . . . . . . . . 10 𝑥((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧)
44 simpr 488 . . . . . . . . . . . 12 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧)
4544eleq2d 2837 . . . . . . . . . . 11 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (𝑦𝑣𝑦𝑧))
4644eleq1d 2836 . . . . . . . . . . 11 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (𝑣𝑤𝑧𝑤))
4745, 46anbi12d 633 . . . . . . . . . 10 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → ((𝑦𝑣𝑣𝑤) ↔ (𝑦𝑧𝑧𝑤)))
4843, 47albid 2222 . . . . . . . . 9 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (∀𝑥(𝑦𝑣𝑣𝑤) ↔ ∀𝑥(𝑦𝑧𝑧𝑤)))
49 nfcvd 2920 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑤𝑣)
50 nfcvf2 2946 . . . . . . . . . . . . 13 (¬ ∀𝑧 𝑧 = 𝑤𝑤𝑧)
51503ad2ant3 1132 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑤𝑧)
5249, 51nfeqd 2929 . . . . . . . . . . 11 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑤 𝑣 = 𝑧)
5326, 52nfan1 2198 . . . . . . . . . 10 𝑤((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧)
54 nfcvd 2920 . . . . . . . . . . . . 13 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑦𝑣)
55 nfcvf2 2946 . . . . . . . . . . . . . 14 (¬ ∀𝑧 𝑧 = 𝑦𝑦𝑧)
56553ad2ant2 1131 . . . . . . . . . . . . 13 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → 𝑦𝑧)
5754, 56nfeqd 2929 . . . . . . . . . . . 12 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → Ⅎ𝑦 𝑣 = 𝑧)
589, 57nfan1 2198 . . . . . . . . . . 11 𝑦((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧)
5947anbi1d 632 . . . . . . . . . . . . 13 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥))))
6053, 59exbid 2223 . . . . . . . . . . . 12 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ ∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥))))
6160bibi1d 347 . . . . . . . . . . 11 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → ((∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ (∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
6258, 61albid 2222 . . . . . . . . . 10 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (∀𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ∀𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
6353, 62exbid 2223 . . . . . . . . 9 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → (∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤) ↔ ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
6448, 63imbi12d 348 . . . . . . . 8 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) ∧ 𝑣 = 𝑧) → ((∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ (∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))))
6564ex 416 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → (𝑣 = 𝑧 → ((∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ (∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))))
6613, 38, 65cbvald 2417 . . . . . 6 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → (∀𝑣(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∀𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))))
679, 66albid 2222 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → (∀𝑦𝑣(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))))
685, 67exbid 2223 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → (∃𝑥𝑦𝑣(∀𝑥(𝑦𝑣𝑣𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑣𝑣𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))))
691, 68mpbii 236 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦 ∧ ¬ ∀𝑧 𝑧 = 𝑤) → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
70693exp 1116 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (¬ ∀𝑧 𝑧 = 𝑤 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))))
71 axacndlem2 10081 . . 3 (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
7271aecoms 2439 . 2 (∀𝑧 𝑧 = 𝑥 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
73 axacndlem3 10082 . . 3 (∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
7473aecoms 2439 . 2 (∀𝑧 𝑧 = 𝑦 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
75 nfae 2444 . . . 4 𝑦𝑧 𝑧 = 𝑤
76 simpr 488 . . . . . . 7 ((𝑦𝑧𝑧𝑤) → 𝑧𝑤)
7776alimi 1813 . . . . . 6 (∀𝑥(𝑦𝑧𝑧𝑤) → ∀𝑥 𝑧𝑤)
78 nd3 10062 . . . . . . 7 (∀𝑧 𝑧 = 𝑤 → ¬ ∀𝑥 𝑧𝑤)
7978pm2.21d 121 . . . . . 6 (∀𝑧 𝑧 = 𝑤 → (∀𝑥 𝑧𝑤 → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
8077, 79syl5 34 . . . . 5 (∀𝑧 𝑧 = 𝑤 → (∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
8180axc4i 2330 . . . 4 (∀𝑧 𝑧 = 𝑤 → ∀𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
8275, 81alrimi 2211 . . 3 (∀𝑧 𝑧 = 𝑤 → ∀𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
838219.8ad 2179 . 2 (∀𝑧 𝑧 = 𝑤 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))
8470, 72, 74, 83pm2.61iii 188 1 𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536  ∃wex 1781  Ⅎwnfc 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-reg 9102  ax-ac 9932 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-eprel 5439  df-fr 5487 This theorem is referenced by:  zfcndac  10092  axacprim  33177
 Copyright terms: Public domain W3C validator