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Theorem axunndlem1 10005
Description: Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
axunndlem1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

Proof of Theorem axunndlem1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 en2lp 9057 . . . . . . . 8 ¬ (𝑦𝑥𝑥𝑦)
2 elequ2 2120 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
32anbi2d 628 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑦𝑥𝑥𝑦) ↔ (𝑦𝑥𝑥𝑧)))
41, 3mtbii 327 . . . . . . 7 (𝑦 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
54sps 2174 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
65nexdv 1928 . . . . 5 (∀𝑦 𝑦 = 𝑧 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
76pm2.21d 121 . . . 4 (∀𝑦 𝑦 = 𝑧 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
87axc4i 2332 . . 3 (∀𝑦 𝑦 = 𝑧 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
9819.8ad 2171 . 2 (∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
10 zfun 7451 . . 3 𝑥𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥)
11 nfnae 2448 . . . . 5 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
12 nfnae 2448 . . . . . . 7 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
13 nfvd 1907 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤𝑥)
14 nfcvf 3004 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
1514nfcrd 2966 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥𝑧)
1613, 15nfand 1889 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑤𝑥𝑥𝑧))
1712, 16nfexd 2339 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑥(𝑤𝑥𝑥𝑧))
1817, 13nfimd 1886 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥))
19 elequ1 2112 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
2019anbi1d 629 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑤𝑥𝑥𝑧) ↔ (𝑦𝑥𝑥𝑧)))
2120exbidv 1913 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑥(𝑤𝑥𝑥𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
2221, 19imbi12d 346 . . . . . 6 (𝑤 = 𝑦 → ((∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2322a1i 11 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑦 → ((∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))))
2411, 18, 23cbvald 2419 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2524exbidv 1913 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑥𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2610, 25mpbii 234 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
279, 26pm2.61i 183 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450  ax-reg 9044
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-eprel 5458  df-fr 5507
This theorem is referenced by:  axunnd  10006
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