 Mathbox for Wolf Lammen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-dveeq12 Structured version   Visualization version   GIF version

Theorem wl-dveeq12 33282
 Description: The current form of ax-13 2244 has a particular disadvantage: The condition ¬ 𝑥 = 𝑦 is less versatile than the general form ¬ ∀𝑥𝑥 = 𝑦. You need ax-10 2017 to arrive at the more general form presented here. You need 19.8a 2050 (or ax-12 2045) to restore 𝑦 = 𝑧 from ∃𝑥𝑦 = 𝑧 again. (Contributed by Wolf Lammen, 9-Jun-2021.)
Assertion
Ref Expression
wl-dveeq12 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem wl-dveeq12
StepHypRef Expression
1 exnal 1752 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2 hbe1 2019 . . 3 (∃𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
3 ax13lem2 2294 . . . . . 6 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
4 ax13lem1 2246 . . . . . 6 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
53, 4syldc 48 . . . . 5 (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
65aleximi 1757 . . . 4 (∀𝑥𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥𝑥 𝑧 = 𝑦))
76com12 32 . . 3 (∃𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝑥 𝑧 = 𝑦 → ∃𝑥𝑥 𝑧 = 𝑦))
8 hbe1a 2020 . . 3 (∃𝑥𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
92, 7, 8syl56 36 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
101, 9sylbir 225 1 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1479  ∃wex 1702 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-13 2244 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator