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Theorem axpowndlem3 9373
Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.)
Assertion
Ref Expression
axpowndlem3 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
Distinct variable group:   𝑦,𝑧

Proof of Theorem axpowndlem3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sp 2051 . . 3 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21con3i 150 . 2 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3 p0ex 4818 . . . . . . . 8 {∅} ∈ V
4 eleq2 2687 . . . . . . . . . 10 (𝑥 = {∅} → (𝑤𝑥𝑤 ∈ {∅}))
54imbi2d 330 . . . . . . . . 9 (𝑥 = {∅} → ((𝑤 = ∅ → 𝑤𝑥) ↔ (𝑤 = ∅ → 𝑤 ∈ {∅})))
65albidv 1846 . . . . . . . 8 (𝑥 = {∅} → (∀𝑤(𝑤 = ∅ → 𝑤𝑥) ↔ ∀𝑤(𝑤 = ∅ → 𝑤 ∈ {∅})))
73, 6spcev 3289 . . . . . . 7 (∀𝑤(𝑤 = ∅ → 𝑤 ∈ {∅}) → ∃𝑥𝑤(𝑤 = ∅ → 𝑤𝑥))
8 0ex 4755 . . . . . . . . 9 ∅ ∈ V
98snid 4184 . . . . . . . 8 ∅ ∈ {∅}
10 eleq1 2686 . . . . . . . 8 (𝑤 = ∅ → (𝑤 ∈ {∅} ↔ ∅ ∈ {∅}))
119, 10mpbiri 248 . . . . . . 7 (𝑤 = ∅ → 𝑤 ∈ {∅})
127, 11mpg 1721 . . . . . 6 𝑥𝑤(𝑤 = ∅ → 𝑤𝑥)
13 neq0 3911 . . . . . . . . . 10 𝑤 = ∅ ↔ ∃𝑥 𝑥𝑤)
1413con1bii 346 . . . . . . . . 9 (¬ ∃𝑥 𝑥𝑤𝑤 = ∅)
1514imbi1i 339 . . . . . . . 8 ((¬ ∃𝑥 𝑥𝑤𝑤𝑥) ↔ (𝑤 = ∅ → 𝑤𝑥))
1615albii 1744 . . . . . . 7 (∀𝑤(¬ ∃𝑥 𝑥𝑤𝑤𝑥) ↔ ∀𝑤(𝑤 = ∅ → 𝑤𝑥))
1716exbii 1771 . . . . . 6 (∃𝑥𝑤(¬ ∃𝑥 𝑥𝑤𝑤𝑥) ↔ ∃𝑥𝑤(𝑤 = ∅ → 𝑤𝑥))
1812, 17mpbir 221 . . . . 5 𝑥𝑤(¬ ∃𝑥 𝑥𝑤𝑤𝑥)
19 nfnae 2317 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
20 nfnae 2317 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
21 nfcvf2 2785 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
22 nfcvd 2762 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑤)
2321, 22nfeld 2769 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥𝑤)
2419, 23nfexd 2164 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥 𝑥𝑤)
2524nfnd 1782 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 ¬ ∃𝑥 𝑥𝑤)
2622, 21nfeld 2769 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤𝑥)
2725, 26nfimd 1820 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(¬ ∃𝑥 𝑥𝑤𝑤𝑥))
28 nfeqf2 2296 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 = 𝑦)
2919, 28nfan1 2066 . . . . . . . . . . 11 𝑥(¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦)
30 elequ2 2001 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
3130adantl 482 . . . . . . . . . . 11 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (𝑥𝑤𝑥𝑦))
3229, 31exbid 2089 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (∃𝑥 𝑥𝑤 ↔ ∃𝑥 𝑥𝑦))
3332notbid 308 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (¬ ∃𝑥 𝑥𝑤 ↔ ¬ ∃𝑥 𝑥𝑦))
34 elequ1 1994 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
3534adantl 482 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → (𝑤𝑥𝑦𝑥))
3633, 35imbi12d 334 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑦) → ((¬ ∃𝑥 𝑥𝑤𝑤𝑥) ↔ (¬ ∃𝑥 𝑥𝑦𝑦𝑥)))
3736ex 450 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → ((¬ ∃𝑥 𝑥𝑤𝑤𝑥) ↔ (¬ ∃𝑥 𝑥𝑦𝑦𝑥))))
3820, 27, 37cbvald 2276 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑤(¬ ∃𝑥 𝑥𝑤𝑤𝑥) ↔ ∀𝑦(¬ ∃𝑥 𝑥𝑦𝑦𝑥)))
3919, 38exbid 2089 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑤(¬ ∃𝑥 𝑥𝑤𝑤𝑥) ↔ ∃𝑥𝑦(¬ ∃𝑥 𝑥𝑦𝑦𝑥)))
4018, 39mpbii 223 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(¬ ∃𝑥 𝑥𝑦𝑦𝑥))
41 nfae 2315 . . . . 5 𝑥𝑥 𝑥 = 𝑧
42 nfae 2315 . . . . . 6 𝑦𝑥 𝑥 = 𝑧
43 axc11r 2186 . . . . . . . . . 10 (∀𝑥 𝑥 = 𝑧 → (∀𝑧 ¬ 𝑥𝑦 → ∀𝑥 ¬ 𝑥𝑦))
44 alnex 1703 . . . . . . . . . 10 (∀𝑧 ¬ 𝑥𝑦 ↔ ¬ ∃𝑧 𝑥𝑦)
45 alnex 1703 . . . . . . . . . 10 (∀𝑥 ¬ 𝑥𝑦 ↔ ¬ ∃𝑥 𝑥𝑦)
4643, 44, 453imtr3g 284 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑧 → (¬ ∃𝑧 𝑥𝑦 → ¬ ∃𝑥 𝑥𝑦))
47 nd3 9363 . . . . . . . . . 10 (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑦 𝑥𝑧)
4847pm2.21d 118 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑥𝑧 → ¬ ∃𝑥 𝑥𝑦))
4946, 48jad 174 . . . . . . . 8 (∀𝑥 𝑥 = 𝑧 → ((∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → ¬ ∃𝑥 𝑥𝑦))
5049spsd 2055 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → ¬ ∃𝑥 𝑥𝑦))
5150imim1d 82 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ((¬ ∃𝑥 𝑥𝑦𝑦𝑥) → (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5242, 51alimd 2079 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∀𝑦(¬ ∃𝑥 𝑥𝑦𝑦𝑥) → ∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5341, 52eximd 2083 . . . 4 (∀𝑥 𝑥 = 𝑧 → (∃𝑥𝑦(¬ ∃𝑥 𝑥𝑦𝑦𝑥) → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5440, 53syl5com 31 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
55 axpowndlem2 9372 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5654, 55pm2.61d 170 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
572, 56syl 17 1 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  c0 3896  {csn 4153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-reg 8449
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156
This theorem is referenced by:  axpowndlem4  9374
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