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Theorem axc11n11r 32315
Description: Proof of axc11n 2306 from { ax-1 6-- ax-7 1932, axc9 2301, axc11r 2186 } (note that axc16 2131 is provable from { ax-1 6-- ax-7 1932, axc11r 2186 }).

Note that axc11n 2306 proves (over minimal calculus) that axc11 2313 and axc11r 2186 are equivalent. Therefore, axc11n11 32314 and axc11n11r 32315 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2313 appears slightly stronger since axc11n11r 32315 requires axc9 2301 while axc11n11 32314 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 1941 . . . . 5 (𝑥 = 𝑦𝑦 = 𝑥)
2 axc16 2131 . . . . 5 (∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
31, 2syl5 34 . . . 4 (∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
43spsd 2055 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
54exlimiv 1855 . 2 (∃𝑧𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
6 alnex 1703 . . 3 (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ↔ ¬ ∃𝑧𝑦 𝑦 = 𝑧)
7 ax6evr 1939 . . . . 5 𝑧 𝑥 = 𝑧
8 19.29 1798 . . . . 5 ((∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ∧ ∃𝑧 𝑥 = 𝑧) → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧))
97, 8mpan2 706 . . . 4 (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧))
10 axc9 2301 . . . . . . . . . . . 12 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)))
1110impcom 446 . . . . . . . . . . 11 ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
12 axc11r 2186 . . . . . . . . . . 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 1980 . . . . . . . . . 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 450 . . . . . . . 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 445 . . . . . 6 ((¬ ∀𝑦 𝑦 = 𝑧𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)))
19 pm2.18 122 . . . . . 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 1855 . . . 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 225 . 2 (¬ ∃𝑧𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
245, 23pm2.61i 176 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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