Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axc11n11r Structured version   Visualization version   GIF version

Theorem axc11n11r 33914
Description: Proof of axc11n 2440 from { ax-1 6-- ax-7 2006, axc9 2391, axc11r 2377 } (note that axc16 2252 is provable from { ax-1 6-- ax-7 2006, axc11r 2377 }).

Note that axc11n 2440 proves (over minimal calculus) that axc11 2444 and axc11r 2377 are equivalent. Therefore, axc11n11 33913 and axc11n11r 33914 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2444 appears slightly stronger since axc11n11r 33914 requires axc9 2391 while axc11n11 33913 does not).

(Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)

Assertion
Ref Expression
axc11n11r (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem axc11n11r
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equcomi 2015 . . . . 5 (𝑥 = 𝑦𝑦 = 𝑥)
2 axc16 2252 . . . . 5 (∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
31, 2syl5 34 . . . 4 (∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
43spsd 2176 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
54exlimiv 1922 . 2 (∃𝑧𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
6 alnex 1773 . . 3 (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ↔ ¬ ∃𝑧𝑦 𝑦 = 𝑧)
7 ax6evr 2013 . . . . 5 𝑧 𝑥 = 𝑧
8 19.29 1865 . . . . 5 ((∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ∧ ∃𝑧 𝑥 = 𝑧) → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧))
97, 8mpan2 687 . . . 4 (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧))
10 axc9 2391 . . . . . . . . . . . 12 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)))
1110impcom 408 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
12 axc11r 2377 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
1311, 12syl9 77 . . . . . . . . . 10 ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧)))
14 aev 2053 . . . . . . . . . 10 (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)
1513, 14syl8 76 . . . . . . . . 9 ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)))
1615ex 413 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))))
1716com24 95 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))))
1817imp 407 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)))
19 pm2.18 128 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥) → ∀𝑦 𝑦 = 𝑥)
2018, 19syl6 35 . . . . 5 ((¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
2120exlimiv 1922 . . . 4 (∃𝑧(¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
229, 21syl 17 . . 3 (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
236, 22sylbir 236 . 2 (¬ ∃𝑧𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
245, 23pm2.61i 183 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator