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

Theorem bj-cleqhyp 33017
 Description: The hypothesis of bj-df-cleq 33018. Note that the hypothesis of bj-df-cleq 33018 actually has an additional dv condition on 𝑥, 𝑦 and therefore is provable by simply using ax-ext 2631 in place of axext3 2633 in the current proof. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-cleqhyp (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-cleqhyp
StepHypRef Expression
1 bj-elequ2g 32791 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
2 axext3 2633 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
31, 2impbii 199 1 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   ↔ 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  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by:  bj-dfcleq  33019
 Copyright terms: Public domain W3C validator