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Theorem axregndlem2 9626
Description: Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
axregndlem2 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Distinct variable group:   𝑦,𝑧

Proof of Theorem axregndlem2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 axreg2 8653 . . . . . 6 (𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
21ax-gen 1869 . . . . 5 𝑤(𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
3 nfnae 2469 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4 nfnae 2469 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
53, 4nfan 1979 . . . . . 6 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
6 nfcvd 2913 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑤)
7 nfcvf 2936 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
87adantr 466 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑦)
96, 8nfeld 2921 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤𝑦)
10 nfv 1994 . . . . . . . 8 𝑤(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
11 nfnae 2469 . . . . . . . . . . 11 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
12 nfnae 2469 . . . . . . . . . . 11 𝑧 ¬ ∀𝑥 𝑥 = 𝑧
1311, 12nfan 1979 . . . . . . . . . 10 𝑧(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
14 nfcvf 2936 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑧)
1514adantl 467 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑧)
1615, 6nfeld 2921 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧𝑤)
1715, 8nfeld 2921 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧𝑦)
1817nfnd 1935 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 ¬ 𝑧𝑦)
1916, 18nfimd 1972 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧𝑤 → ¬ 𝑧𝑦))
2013, 19nfald 2326 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑧(𝑧𝑤 → ¬ 𝑧𝑦))
219, 20nfand 1977 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
2210, 21nfexd 2328 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
239, 22nfimd 1972 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))))
24 simpr 471 . . . . . . . . 9 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → 𝑤 = 𝑥)
2524eleq1d 2834 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑤𝑦𝑥𝑦))
26 nfcvd 2913 . . . . . . . . . . . . . . 15 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑧𝑤)
27 nfcvf2 2937 . . . . . . . . . . . . . . . 16 (¬ ∀𝑥 𝑥 = 𝑧𝑧𝑥)
2827adantl 467 . . . . . . . . . . . . . . 15 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑧𝑥)
2926, 28nfeqd 2920 . . . . . . . . . . . . . 14 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑥)
3013, 29nfan1 2221 . . . . . . . . . . . . 13 𝑧((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
3124eleq2d 2835 . . . . . . . . . . . . . 14 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑧𝑤𝑧𝑥))
3231imbi1d 330 . . . . . . . . . . . . 13 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑧𝑤 → ¬ 𝑧𝑦) ↔ (𝑧𝑥 → ¬ 𝑧𝑦)))
3330, 32albid 2245 . . . . . . . . . . . 12 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦) ↔ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
3425, 33anbi12d 608 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3534ex 397 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
365, 21, 35cbvexd 2436 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3736adantr 466 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3825, 37imbi12d 333 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))) ↔ (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
3938ex 397 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))) ↔ (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))))
405, 23, 39cbvald 2435 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑤(𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))) ↔ ∀𝑥(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
412, 40mpbii 223 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∀𝑥(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
424119.21bi 2212 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
4342ex 397 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
44 elirrv 8656 . . . . 5 ¬ 𝑥𝑥
45 elequ2 2158 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
4644, 45mtbii 315 . . . 4 (𝑥 = 𝑦 → ¬ 𝑥𝑦)
4746sps 2208 . . 3 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
4847pm2.21d 119 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
49 axregndlem1 9625 . 2 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
5043, 48, 49pm2.61ii 177 1 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1628  wex 1851  wnfc 2899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-reg 8652
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-dif 3724  df-un 3726  df-nul 4062  df-sn 4315  df-pr 4317
This theorem is referenced by:  axregnd  9627
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