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Theorem dtruALT 5257
Description: Alternate proof of dtru 5239 which requires more axioms but is shorter and may be easier to understand.

Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that 𝑥 and 𝑦 be distinct. Specifically, theorem spcev 3558 requires that 𝑥 must not occur in the subexpression ¬ 𝑦 = {∅} in step 4 nor in the subexpression ¬ 𝑦 = ∅ in step 9. The proof verifier will require that 𝑥 and 𝑦 be in a distinct variable group to ensure this. You can check this by deleting the $d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
dtruALT ¬ ∀𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruALT
StepHypRef Expression
1 0inp0 5227 . . . 4 (𝑦 = ∅ → ¬ 𝑦 = {∅})
2 p0ex 5253 . . . . 5 {∅} ∈ V
3 eqeq2 2813 . . . . . 6 (𝑥 = {∅} → (𝑦 = 𝑥𝑦 = {∅}))
43notbid 321 . . . . 5 (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅}))
52, 4spcev 3558 . . . 4 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥)
61, 5syl 17 . . 3 (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
7 0ex 5178 . . . 4 ∅ ∈ V
8 eqeq2 2813 . . . . 5 (𝑥 = ∅ → (𝑦 = 𝑥𝑦 = ∅))
98notbid 321 . . . 4 (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅))
107, 9spcev 3558 . . 3 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
116, 10pm2.61i 185 . 2 𝑥 ¬ 𝑦 = 𝑥
12 exnal 1828 . . 3 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥)
13 eqcom 2808 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
1413albii 1821 . . 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 1536   = wceq 1538  wex 1781  c0 4246  {csn 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529
This theorem is referenced by: (None)
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