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Theorem axunndlem1 10590
Description: Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
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 9601 . . . . . . . 8 ¬ (𝑦𝑥𝑥𝑦)
2 elequ2 2122 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
32anbi2d 630 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑦𝑥𝑥𝑦) ↔ (𝑦𝑥𝑥𝑧)))
41, 3mtbii 326 . . . . . . 7 (𝑦 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
54sps 2179 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
65nexdv 1940 . . . . 5 (∀𝑦 𝑦 = 𝑧 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
76pm2.21d 121 . . . 4 (∀𝑦 𝑦 = 𝑧 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
87axc4i 2316 . . 3 (∀𝑦 𝑦 = 𝑧 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
9819.8ad 2176 . 2 (∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
10 zfun 7726 . . 3 𝑥𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥)
11 nfnae 2434 . . . . 5 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
12 nfnae 2434 . . . . . . 7 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
13 nfvd 1919 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤𝑥)
14 nfcvf 2933 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
1514nfcrd 2893 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥𝑧)
1613, 15nfand 1901 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑤𝑥𝑥𝑧))
1712, 16nfexd 2323 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑥(𝑤𝑥𝑥𝑧))
1817, 13nfimd 1898 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥))
19 elequ1 2114 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
2019anbi1d 631 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑤𝑥𝑥𝑧) ↔ (𝑦𝑥𝑥𝑧)))
2120exbidv 1925 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑥(𝑤𝑥𝑥𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
2221, 19imbi12d 345 . . . . . 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 2407 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2524exbidv 1925 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑥𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2610, 25mpbii 232 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
279, 26pm2.61i 182 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2372  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-reg 9587
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-fr 5632
This theorem is referenced by:  axunnd  10591
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