Theorem axc11n11r 35148

Description: Proof of axc11n 2424 from { ax-1 6-- ax-7 2011, axc9 2380, axc11r 2364 } (note that axc16 2252 is provable from { ax-1 6-- ax-7 2011, axc11r 2364 }).

Note that axc11n 2424 proves (over minimal calculus) that axc11 2428 and axc11r 2364 are equivalent. Therefore, axc11n11 35147 and axc11n11r 35148 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2428 appears slightly stronger since axc11n11r 35148 requires axc9 2380 while axc11n11 35147 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 2020	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
2		axc16 2252	. . . . 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 2180	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
5	4	exlimiv 1933	. 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 2018	. . . . 5 ⊢ ∃𝑧 𝑥 = 𝑧
8		19.29 1876	. . . . 5 ⊢ ((∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ∧ ∃𝑧 𝑥 = 𝑧) → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧))
9	7, 8	mpan2 689	. . . 4 ⊢ (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧))
10		axc9 2380	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)))
11	10	impcom 408	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
12		axc11r 2364	. . . . . . . . . . 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 2060	. . . . . . . . . 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 413	. . . . . . . 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 407	. . . . . 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 1933	. . . 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 234	. 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 396  ∀wal 1539  ∃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 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2370

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)