Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-elequ2g Structured version   Visualization version   GIF version

Theorem bj-elequ2g 32791
Description: A form of elequ2 2044 with a universal quantifier. Its converse is ax-ext 2631. (TODO: move to main part, minimize axext4 2635--- as of 4-Nov-2020, minimizes only axext4 2635, by 13 bytes; and link to it in the comment of ax-ext 2631.) (Contributed by BJ, 3-Oct-2019.)
Assertion
Ref Expression
bj-elequ2g (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-elequ2g
StepHypRef Expression
1 elequ2 2044 . 2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21alrimiv 1895 1 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521
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-9 2039
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  bj-axext4  32895  bj-cleqhyp  33017
  Copyright terms: Public domain W3C validator