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Theorem dtruALT2 5375
Description: Alternate proof of dtru 5446 using ax-pow 5370 instead of ax-pr 5437. See dtruALT 5393 for another proof using ax-pow 5370 instead of ax-pr 5437. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2374. (Revised by BJ, 31-May-2019.) Avoid ax-12 2174. (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 5374 . . . 4 𝑤 𝑥𝑤
2 ax-nul 5311 . . . . 5 𝑧𝑥 ¬ 𝑥𝑧
3 elequ1 2112 . . . . . . 7 (𝑥 = 𝑤 → (𝑥𝑧𝑤𝑧))
43notbid 318 . . . . . 6 (𝑥 = 𝑤 → (¬ 𝑥𝑧 ↔ ¬ 𝑤𝑧))
54spw 2030 . . . . 5 (∀𝑥 ¬ 𝑥𝑧 → ¬ 𝑥𝑧)
62, 5eximii 1833 . . . 4 𝑧 ¬ 𝑥𝑧
7 exdistrv 1952 . . . 4 (∃𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧) ↔ (∃𝑤 𝑥𝑤 ∧ ∃𝑧 ¬ 𝑥𝑧))
81, 6, 7mpbir2an 711 . . 3 𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧)
9 ax9v2 2118 . . . . . 6 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
109com12 32 . . . . 5 (𝑥𝑤 → (𝑤 = 𝑧𝑥𝑧))
1110con3dimp 408 . . . 4 ((𝑥𝑤 ∧ ¬ 𝑥𝑧) → ¬ 𝑤 = 𝑧)
12112eximi 1832 . . 3 (∃𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧) → ∃𝑤𝑧 ¬ 𝑤 = 𝑧)
13 equequ2 2022 . . . . . . 7 (𝑧 = 𝑦 → (𝑤 = 𝑧𝑤 = 𝑦))
1413notbid 318 . . . . . 6 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦))
15 ax7v1 2006 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 = 𝑦𝑤 = 𝑦))
1615con3d 152 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
1716spimevw 1991 . . . . . 6 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
1814, 17biimtrdi 253 . . . . 5 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
19 ax7v1 2006 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
2019con3d 152 . . . . . . 7 (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
2120spimevw 1991 . . . . . 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 1929 . . 3 (∃𝑤𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
258, 12, 24mp2b 10 . 2 𝑥 ¬ 𝑥 = 𝑦
26 exnal 1823 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2725, 26mpbi 230 1 ¬ ∀𝑥 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1534   = wceq 1536  wex 1775  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-nul 5311  ax-pow 5370
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776
This theorem is referenced by:  dtrucor  5376  dvdemo1  5378  nfnid  5380  axc16b  5394  eunex  5395  brprcneuALT  6897
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