Theorem axc11n11r 33524

Description: Proof of axc11n 2362 from { ax-1 6-- ax-7 1965, axc9 2312, axc11r 2300 } (note that axc16 2188 is provable from { ax-1 6-- ax-7 1965, axc11r 2300 }).

Note that axc11n 2362 proves (over minimal calculus) that axc11 2366 and axc11r 2300 are equivalent. Therefore, axc11n11 33523 and axc11n11r 33524 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2366 appears slightly stronger since axc11n11r 33524 requires axc9 2312 while axc11n11 33523 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 1974	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
2		axc16 2188	. . . . 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 2115	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
5	4	exlimiv 1889	. 2 ⊢ (∃𝑧∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
6		alnex 1744	. . 3 ⊢ (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ↔ ¬ ∃𝑧∀𝑦 𝑦 = 𝑧)
7		ax6evr 1972	. . . . 5 ⊢ ∃𝑧 𝑥 = 𝑧
8		19.29 1836	. . . . 5 ⊢ ((∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 ∧ ∃𝑧 𝑥 = 𝑧) → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧))
9	7, 8	mpan2 678	. . . 4 ⊢ (∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑧(¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧))
10		axc9 2312	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)))
11	10	impcom 399	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
12		axc11r 2300	. . . . . . . . . . 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 2006	. . . . . . . . . 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 405	. . . . . . . 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 398	. . . . . 6 ⊢ ((¬ ∀𝑦 𝑦 = 𝑧 ∧ 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)))
19		pm2.18 125	. . . . . 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 1889	. . . 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 227	. 2 ⊢ (¬ ∃𝑧∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥))
24	5, 23	pm2.61i 177	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387  ∀wal 1505  ∃wex 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106  ax-13 2301

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747

This theorem is referenced by: (None)