Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-hbaeb2 Structured version   Visualization version   GIF version

Theorem bj-hbaeb2 31993
 Description: Biconditional version of a form of hbae 2303 with commuted quantifiers, not requiring ax-11 2021. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-hbaeb2 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)

Proof of Theorem bj-hbaeb2
StepHypRef Expression
1 sp 2041 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2 axc9 2290 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
31, 2syl7 72 . . . 4 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
4 axc11r 2175 . . . 4 (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
5 axc11 2302 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
65pm2.43i 50 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
7 axc11r 2175 . . . . 5 (∀𝑧 𝑧 = 𝑦 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
86, 7syl5 33 . . . 4 (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
93, 4, 8pm2.61ii 176 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
109axc4i 2116 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑧 𝑥 = 𝑦)
11 sp 2041 . . 3 (∀𝑧 𝑥 = 𝑦𝑥 = 𝑦)
1211alimi 1730 . 2 (∀𝑥𝑧 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
1310, 12impbii 198 1 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  bj-hbaeb  31994  bj-dvv  31996
 Copyright terms: Public domain W3C validator