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Theorem dtruALT2 5293
Description: Alternate proof of dtru 5359 using ax-pow 5288 instead of ax-pr 5352. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2372. (Revised by Gino Giotto, 5-Sep-2023.) Avoid ax-12 2171. (Revised by Rohan Ridenour, 9-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dtruALT2 ¬ ∀𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruALT2
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elALT2 5292 . . . 4 𝑤 𝑥𝑤
2 ax-nul 5230 . . . . 5 𝑧𝑥 ¬ 𝑥𝑧
3 elequ1 2113 . . . . . . 7 (𝑥 = 𝑤 → (𝑥𝑧𝑤𝑧))
43notbid 318 . . . . . 6 (𝑥 = 𝑤 → (¬ 𝑥𝑧 ↔ ¬ 𝑤𝑧))
54spw 2037 . . . . 5 (∀𝑥 ¬ 𝑥𝑧 → ¬ 𝑥𝑧)
62, 5eximii 1839 . . . 4 𝑧 ¬ 𝑥𝑧
7 exdistrv 1959 . . . 4 (∃𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧) ↔ (∃𝑤 𝑥𝑤 ∧ ∃𝑧 ¬ 𝑥𝑧))
81, 6, 7mpbir2an 708 . . 3 𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧)
9 ax9v2 2119 . . . . . 6 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
109com12 32 . . . . 5 (𝑥𝑤 → (𝑤 = 𝑧𝑥𝑧))
1110con3dimp 409 . . . 4 ((𝑥𝑤 ∧ ¬ 𝑥𝑧) → ¬ 𝑤 = 𝑧)
12112eximi 1838 . . 3 (∃𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧) → ∃𝑤𝑧 ¬ 𝑤 = 𝑧)
13 equequ2 2029 . . . . . . 7 (𝑧 = 𝑦 → (𝑤 = 𝑧𝑤 = 𝑦))
1413notbid 318 . . . . . 6 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦))
15 ax7v1 2013 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 = 𝑦𝑤 = 𝑦))
1615con3d 152 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
1716spimevw 1998 . . . . . 6 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
1814, 17syl6bi 252 . . . . 5 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
19 ax7v1 2013 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
2019con3d 152 . . . . . . 7 (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
2120spimevw 1998 . . . . . 6 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
2221a1d 25 . . . . 5 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
2318, 22pm2.61i 182 . . . 4 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
2423exlimivv 1935 . . 3 (∃𝑤𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
258, 12, 24mp2b 10 . 2 𝑥 ¬ 𝑥 = 𝑦
26 exnal 1829 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2725, 26mpbi 229 1 ¬ ∀𝑥 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1537   = wceq 1539  wex 1782  wcel 2106
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-nul 5230  ax-pow 5288
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783
This theorem is referenced by:  dtrucor  5294  dvdemo1  5296  nfnid  5298  axc16b  5312  eunex  5313  brprcneuALT  6765
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