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Theorem dtruALT2 4838
 Description: Alternate proof of dtru 4783 using ax-pr 4833 instead of ax-pow 4769. (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 4763 . . . 4 (𝑦 = ∅ → ¬ 𝑦 = {∅})
2 snex 4835 . . . . 5 {∅} ∈ V
3 eqeq2 2621 . . . . . 6 (𝑥 = {∅} → (𝑦 = 𝑥𝑦 = {∅}))
43notbid 307 . . . . 5 (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅}))
52, 4spcev 3273 . . . 4 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥)
61, 5syl 17 . . 3 (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
7 0ex 4718 . . . 4 ∅ ∈ V
8 eqeq2 2621 . . . . 5 (𝑥 = ∅ → (𝑦 = 𝑥𝑦 = ∅))
98notbid 307 . . . 4 (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅))
107, 9spcev 3273 . . 3 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
116, 10pm2.61i 175 . 2 𝑥 ¬ 𝑦 = 𝑥
12 exnal 1744 . . 3 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥)
13 eqcom 2617 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
1413albii 1737 . . 3 (∀𝑥 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
1512, 14xchbinx 323 . 2 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
1611, 15mpbi 219 1 ¬ ∀𝑥 𝑥 = 𝑦
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∀wal 1473   = wceq 1475  ∃wex 1695  ∅c0 3874  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128 This theorem is referenced by: (None)
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