MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax6vsep Structured version   Visualization version   GIF version

Theorem ax6vsep 4750
Description: Derive ax6v 1886 (a weakened version of ax-6 1885 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 4746 and Extensionality ax-ext 2601. See ax6 2250 for the derivation of ax-6 1885 from ax6v 1886. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6vsep ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem ax6vsep
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax-sep 4746 . . 3 𝑥𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧)))
2 id 22 . . . . . . . . 9 (𝑧 = 𝑧𝑧 = 𝑧)
32biantru 526 . . . . . . . 8 (𝑧𝑦 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧)))
43bibi2i 327 . . . . . . 7 ((𝑧𝑥𝑧𝑦) ↔ (𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))))
54biimpri 218 . . . . . 6 ((𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → (𝑧𝑥𝑧𝑦))
65alimi 1736 . . . . 5 (∀𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → ∀𝑧(𝑧𝑥𝑧𝑦))
7 ax-ext 2601 . . . . 5 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
86, 7syl 17 . . . 4 (∀𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → 𝑥 = 𝑦)
98eximi 1759 . . 3 (∃𝑥𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → ∃𝑥 𝑥 = 𝑦)
101, 9ax-mp 5 . 2 𝑥 𝑥 = 𝑦
11 df-ex 1702 . 2 (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
1210, 11mpbi 220 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-ext 2601  ax-sep 4746
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator