Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dveeq1-o16 Structured version   Visualization version   GIF version

Theorem dveeq1-o16 36687
Description: Version of dveeq1 2379 using ax-c16 36643 instead of ax-5 1918. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1918. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq1-o16 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq1-o16
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax5eq 36683 . 2 (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2 ax5eq 36683 . 2 (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3 equequ1 2033 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
41, 2, 3dvelimh 2449 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371  ax-c9 36641  ax-c16 36643
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator