Theorem axc11n11r 36928

Description: Proof of axc11n 2431 from { ax-1 6-- ax-7 2010, axc9 2387, axc11r 2373 } (note that axc16 2269 is provable from { ax-1 6-- ax-7 2010, axc11r 2373 }).

Note that axc11n 2431 proves (over minimal calculus) that axc11 2435 and axc11r 2373 are equivalent. Therefore, axc11n11 36927 and axc11n11r 36928 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2435 appears slightly stronger since axc11n11r 36928 requires axc9 2387 while axc11n11 36927 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.

Step	Hyp	Ref	Expression
1		equcomi 2019	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
2		axc16 2269	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
3	1, 2	syl5 34	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
4	3	spsd 2195	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
5	4	exlimiv 1932	. 2 ⊢ (∃𝑧∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
6		alnex 1783	. . 3 ⊢ (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ↔ ¬ ∃𝑧∀𝑦 𝑦 = 𝑧)
7		ax6evr 2017	. . . . 5 ⊢ ∃𝑧 𝑥 = 𝑧
8		19.29 1875	. . . . 5 ⊢ ((∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ∧ ∃𝑧 𝑥 = 𝑧) → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧))
9	7, 8	mpan2 692	. . . 4 ⊢ (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧))
10		axc9 2387	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)))
11	10	impcom 407	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
12		axc11r 2373	. . . . . . . . . . 11 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
13	11, 12	syl9 77	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧)))
14		aev 2061	. . . . . . . . . 10 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)
15	13, 14	syl8 76	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)))
16	15	ex 412	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))))
17	16	com24 95	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))))
18	17	imp 406	. . . . . 6 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)))
19		pm2.18 128	. . . . . 6 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥) → ∀𝑦 𝑦 = 𝑥)
20	18, 19	syl6 35	. . . . 5 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
21	20	exlimiv 1932	. . . 4 ⊢ (∃𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
22	9, 21	syl 17	. . 3 ⊢ (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
23	6, 22	sylbir 235	. 2 ⊢ (¬ ∃𝑧∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
24	5, 23	pm2.61i 182	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1540  ∃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 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-12 2185  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)