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Theorem axunndlem1 10009
Description: Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2383. (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 9061 . . . . . . . 8 ¬ (𝑦𝑥𝑥𝑦)
2 elequ2 2122 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
32anbi2d 630 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑦𝑥𝑥𝑦) ↔ (𝑦𝑥𝑥𝑧)))
41, 3mtbii 328 . . . . . . 7 (𝑦 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
54sps 2176 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
65nexdv 1930 . . . . 5 (∀𝑦 𝑦 = 𝑧 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
76pm2.21d 121 . . . 4 (∀𝑦 𝑦 = 𝑧 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
87axc4i 2334 . . 3 (∀𝑦 𝑦 = 𝑧 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
9819.8ad 2173 . 2 (∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
10 zfun 7454 . . 3 𝑥𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥)
11 nfnae 2449 . . . . 5 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
12 nfnae 2449 . . . . . . 7 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
13 nfvd 1909 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤𝑥)
14 nfcvf 3005 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑧𝑦𝑧)
1514nfcrd 2967 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥𝑧)
1613, 15nfand 1891 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑤𝑥𝑥𝑧))
1712, 16nfexd 2341 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑥(𝑤𝑥𝑥𝑧))
1817, 13nfimd 1888 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥))
19 elequ1 2114 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
2019anbi1d 631 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑤𝑥𝑥𝑧) ↔ (𝑦𝑥𝑥𝑧)))
2120exbidv 1915 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑥(𝑤𝑥𝑥𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
2221, 19imbi12d 347 . . . . . 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 2421 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2524exbidv 1915 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑥𝑤(∃𝑥(𝑤𝑥𝑥𝑧) → 𝑤𝑥) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
2610, 25mpbii 235 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
279, 26pm2.61i 184 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1528  wex 1773
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 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2383  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453  ax-reg 9048
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-eprel 5458  df-fr 5507
This theorem is referenced by:  axunnd  10010
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