Theorem axc11n11r 36684

Description: Proof of axc11n 2431 from { ax-1 6-- ax-7 2007, axc9 2387, axc11r 2371 } (note that axc16 2261 is provable from { ax-1 6-- ax-7 2007, axc11r 2371 }).

Note that axc11n 2431 proves (over minimal calculus) that axc11 2435 and axc11r 2371 are equivalent. Therefore, axc11n11 36683 and axc11n11r 36684 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 36684 requires axc9 2387 while axc11n11 36683 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 2016	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
2		axc16 2261	. . . . 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 2187	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
5	4	exlimiv 1930	. 2 ⊢ (∃𝑧∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
6		alnex 1781	. . 3 ⊢ (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ↔ ¬ ∃𝑧∀𝑦 𝑦 = 𝑧)
7		ax6evr 2014	. . . . 5 ⊢ ∃𝑧 𝑥 = 𝑧
8		19.29 1873	. . . . 5 ⊢ ((∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ∧ ∃𝑧 𝑥 = 𝑧) → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧))
9	7, 8	mpan2 691	. . . 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 2371	. . . . . . . . . . 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 2057	. . . . . . . . . 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 1930	. . . 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 1538  ∃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 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)