Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-el Structured version   Visualization version   GIF version

Theorem bj-el 32921
 Description: Remove dependency on ax-13 2282 from el 4877. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-el 𝑦 𝑥𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-el
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bj-zfpow 32920 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
2 ax9 2043 . . . . 5 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
32alrimiv 1895 . . . 4 (𝑧 = 𝑥 → ∀𝑦(𝑦𝑧𝑦𝑥))
4 ax8 2036 . . . 4 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
53, 4embantd 59 . . 3 (𝑧 = 𝑥 → ((∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦))
65bj-spimvv 32846 . 2 (∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦)
71, 6eximii 1804 1 𝑦 𝑥𝑦
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-pow 4873 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750 This theorem is referenced by:  bj-dtru  32922  bj-dvdemo2  32928
 Copyright terms: Public domain W3C validator