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Theorem dtruALT2 5303
Description: Alternate proof of dtru 5238 using ax-pr 5297 instead of ax-pow 5233. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dtruALT2 ¬ ∀𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruALT2
StepHypRef Expression
1 0inp0 5226 . . . 4 (𝑦 = ∅ → ¬ 𝑦 = {∅})
2 snex 5299 . . . . 5 {∅} ∈ V
3 eqeq2 2751 . . . . . 6 (𝑥 = {∅} → (𝑦 = 𝑥𝑦 = {∅}))
43notbid 321 . . . . 5 (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅}))
52, 4spcev 3511 . . . 4 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥)
61, 5syl 17 . . 3 (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
7 0ex 5176 . . . 4 ∅ ∈ V
8 eqeq2 2751 . . . . 5 (𝑥 = ∅ → (𝑦 = 𝑥𝑦 = ∅))
98notbid 321 . . . 4 (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅))
107, 9spcev 3511 . . 3 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
116, 10pm2.61i 185 . 2 𝑥 ¬ 𝑦 = 𝑥
12 exnal 1833 . . 3 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥)
13 eqcom 2746 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
1413albii 1826 . . 3 (∀𝑥 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
1512, 14xchbinx 337 . 2 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
1611, 15mpbi 233 1 ¬ ∀𝑥 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1540   = wceq 1542  wex 1786  c0 4212  {csn 4517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-v 3401  df-dif 3847  df-un 3849  df-nul 4213  df-sn 4518  df-pr 4520
This theorem is referenced by:  fvprc  6669
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