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Theorem axc11n11r 34197
 Description: Proof of axc11n 2437 from { ax-1 6-- ax-7 2015, axc9 2389, axc11r 2375 } (note that axc16 2259 is provable from { ax-1 6-- ax-7 2015, axc11r 2375 }). Note that axc11n 2437 proves (over minimal calculus) that axc11 2441 and axc11r 2375 are equivalent. Therefore, axc11n11 34196 and axc11n11r 34197 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2441 appears slightly stronger since axc11n11r 34197 requires axc9 2389 while axc11n11 34196 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 2024 . . . . 5 (𝑥 = 𝑦𝑦 = 𝑥)
2 axc16 2259 . . . . 5 (∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
31, 2syl5 34 . . . 4 (∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
43spsd 2184 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
54exlimiv 1931 . 2 (∃𝑧𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
6 alnex 1783 . . 3 (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ↔ ¬ ∃𝑧𝑦 𝑦 = 𝑧)
7 ax6evr 2022 . . . . 5 𝑧 𝑥 = 𝑧
8 19.29 1874 . . . . 5 ((∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ∧ ∃𝑧 𝑥 = 𝑧) → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧))
97, 8mpan2 690 . . . 4 (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧))
10 axc9 2389 . . . . . . . . . . . 12 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)))
1110impcom 411 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
12 axc11r 2375 . . . . . . . . . . 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 2062 . . . . . . . . . 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 416 . . . . . . . 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 410 . . . . . 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 238 . 2 (¬ ∃𝑧𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
245, 23pm2.61i 185 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by: (None)
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