Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dveel2ALT Structured version   Visualization version   GIF version

Theorem dveel2ALT 34827
Description: Alternate proof of dveel2 2462 using ax-c16 34780 instead of ax-5 2005. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveel2ALT (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel2ALT
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax5el 34825 . 2 (𝑧𝑤 → ∀𝑥 𝑧𝑤)
2 ax5el 34825 . 2 (𝑧𝑦 → ∀𝑤 𝑧𝑦)
3 elequ2 2169 . 2 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
41, 2, 3dvelimh 2430 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-c14 34779  ax-c16 34780
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator