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Theorem axc11n11r 35864
Description: Proof of axc11n 2423 from { ax-1 6-- ax-7 2009, axc9 2379, axc11r 2363 } (note that axc16 2250 is provable from { ax-1 6-- ax-7 2009, axc11r 2363 }).

Note that axc11n 2423 proves (over minimal calculus) that axc11 2427 and axc11r 2363 are equivalent. Therefore, axc11n11 35863 and axc11n11r 35864 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2427 appears slightly stronger since axc11n11r 35864 requires axc9 2379 while axc11n11 35863 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 2018 . . . . 5 (𝑥 = 𝑦𝑦 = 𝑥)
2 axc16 2250 . . . . 5 (∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
31, 2syl5 34 . . . 4 (∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
43spsd 2178 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
54exlimiv 1931 . 2 (∃𝑧𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
6 alnex 1781 . . 3 (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ↔ ¬ ∃𝑧𝑦 𝑦 = 𝑧)
7 ax6evr 2016 . . . . 5 𝑧 𝑥 = 𝑧
8 19.29 1874 . . . . 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 2379 . . . . . . . . . . . 12 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)))
1110impcom 406 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
12 axc11r 2363 . . . . . . . . . . 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 2058 . . . . . . . . . 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 411 . . . . . . . 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 405 . . . . . 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 1931 . . . 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 234 . 2 (¬ ∃𝑧𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
245, 23pm2.61i 182 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wal 1537  wex 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169  ax-13 2369
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784
This theorem is referenced by: (None)
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