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

Theorem bj-zfpow 33131
Description: Remove dependency on ax-13 2408 from zfpow 4975. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-zfpow 𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem bj-zfpow
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-pow 4974 . 2 𝑥𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥)
2 elequ1 2152 . . . . . . 7 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
3 elequ1 2152 . . . . . . 7 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
42, 3imbi12d 333 . . . . . 6 (𝑤 = 𝑥 → ((𝑤𝑦𝑤𝑧) ↔ (𝑥𝑦𝑥𝑧)))
54bj-cbvalvv 33069 . . . . 5 (∀𝑤(𝑤𝑦𝑤𝑧) ↔ ∀𝑥(𝑥𝑦𝑥𝑧))
65imbi1i 338 . . . 4 ((∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥) ↔ (∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥))
76albii 1895 . . 3 (∀𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥) ↔ ∀𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥))
87exbii 1924 . 2 (∃𝑥𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥))
91, 8mpbi 220 1 𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-10 2174  ax-11 2190  ax-12 2203  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-nf 1858
This theorem is referenced by:  bj-el  33132
  Copyright terms: Public domain W3C validator