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Theorem axregndlem2 10017
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 9049 . . . . . 6 (𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
21ax-gen 1789 . . . . 5 𝑤(𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
3 nfnae 2453 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4 nfnae 2453 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
53, 4nfan 1893 . . . . . 6 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
6 nfcvd 2982 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑤)
7 nfcvf 3011 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
87adantr 481 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑦)
96, 8nfeld 2993 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤𝑦)
10 nfv 1908 . . . . . . . 8 𝑤(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
11 nfnae 2453 . . . . . . . . . . 11 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
12 nfnae 2453 . . . . . . . . . . 11 𝑧 ¬ ∀𝑥 𝑥 = 𝑧
1311, 12nfan 1893 . . . . . . . . . 10 𝑧(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
14 nfcvf 3011 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑧)
1514adantl 482 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑧)
1615, 6nfeld 2993 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧𝑤)
1715, 8nfeld 2993 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧𝑦)
1817nfnd 1851 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 ¬ 𝑧𝑦)
1916, 18nfimd 1888 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧𝑤 → ¬ 𝑧𝑦))
2013, 19nfald 2341 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑧(𝑧𝑤 → ¬ 𝑧𝑦))
219, 20nfand 1891 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
2210, 21nfexd 2342 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)))
239, 22nfimd 1888 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))))
24 simpr 485 . . . . . . . . 9 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → 𝑤 = 𝑥)
2524eleq1d 2901 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑤𝑦𝑥𝑦))
26 nfcvd 2982 . . . . . . . . . . . . . . 15 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑧𝑤)
27 nfcvf2 3012 . . . . . . . . . . . . . . . 16 (¬ ∀𝑥 𝑥 = 𝑧𝑧𝑥)
2827adantl 482 . . . . . . . . . . . . . . 15 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑧𝑥)
2926, 28nfeqd 2992 . . . . . . . . . . . . . 14 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑥)
3013, 29nfan1 2191 . . . . . . . . . . . . 13 𝑧((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
3124eleq2d 2902 . . . . . . . . . . . . . 14 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑧𝑤𝑧𝑥))
3231imbi1d 343 . . . . . . . . . . . . 13 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑧𝑤 → ¬ 𝑧𝑦) ↔ (𝑧𝑥 → ¬ 𝑧𝑦)))
3330, 32albid 2215 . . . . . . . . . . . 12 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦) ↔ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
3425, 33anbi12d 630 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3534ex 413 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
365, 21, 35cbvexd 2425 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3736adantr 481 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦)) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
3825, 37imbi12d 346 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))) ↔ (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
3938ex 413 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))) ↔ (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))))
405, 23, 39cbvald 2424 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑤(𝑤𝑦 → ∃𝑤(𝑤𝑦 ∧ ∀𝑧(𝑧𝑤 → ¬ 𝑧𝑦))) ↔ ∀𝑥(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
412, 40mpbii 234 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∀𝑥(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
424119.21bi 2178 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
4342ex 413 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))))
44 elirrv 9052 . . . . 5 ¬ 𝑥𝑥
45 elequ2 2121 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
4644, 45mtbii 327 . . . 4 (𝑥 = 𝑦 → ¬ 𝑥𝑦)
4746sps 2174 . . 3 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
4847pm2.21d 121 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
49 axregndlem1 10016 . 2 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
5043, 48, 49pm2.61ii 184 1 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1528  wex 1773  wnfc 2965
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-reg 9048
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-v 3501  df-dif 3942  df-un 3944  df-nul 4295  df-sn 4564  df-pr 4566
This theorem is referenced by:  axregnd  10018
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