Theorem axc11n-16 38940

Description: This theorem shows that, given ax-c16 38894, we can derive a version of ax-c11n 38890. However, it is weaker than ax-c11n 38890 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc11n-16	⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)

Distinct variable group:   𝑥,𝑧

Proof of Theorem axc11n-16

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-c16 38894	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤))
2	1	alrimiv 1926	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤))
3	2	axc4i-o 38900	. 2 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑥∀𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤))
4		equequ1 2023	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑤 ↔ 𝑧 = 𝑤))
5	4	cbvalvw 2034	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑤 ↔ ∀𝑧 𝑧 = 𝑤)
6	5	a1i 11	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑤 ↔ ∀𝑧 𝑧 = 𝑤))
7	4, 6	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) ↔ (𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤)))
8	7	albidv 1919	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) ↔ ∀𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤)))
9	8	cbvalvw 2034	. . 3 ⊢ (∀𝑥∀𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) ↔ ∀𝑧∀𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
10	9	biimpi 216	. 2 ⊢ (∀𝑥∀𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) → ∀𝑧∀𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
11		nfa1-o 38917	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑧 𝑧 = 𝑤
12	11	19.23 2210	. . . . . 6 ⊢ (∀𝑧(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) ↔ (∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
13	12	albii 1818	. . . . 5 ⊢ (∀𝑤∀𝑧(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) ↔ ∀𝑤(∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
14		ax6ev 1968	. . . . . . . 8 ⊢ ∃𝑧 𝑧 = 𝑤
15		pm2.27 42	. . . . . . . 8 ⊢ (∃𝑧 𝑧 = 𝑤 → ((∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → ∀𝑧 𝑧 = 𝑤))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ ((∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → ∀𝑧 𝑧 = 𝑤)
17	16	alimi 1810	. . . . . 6 ⊢ (∀𝑤(∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → ∀𝑤∀𝑧 𝑧 = 𝑤)
18		equequ2 2024	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑥))
19	18	spv 2397	. . . . . . . 8 ⊢ (∀𝑤 𝑧 = 𝑤 → 𝑧 = 𝑥)
20	19	sps-o 38910	. . . . . . 7 ⊢ (∀𝑧∀𝑤 𝑧 = 𝑤 → 𝑧 = 𝑥)
21	20	alcoms 2157	. . . . . 6 ⊢ (∀𝑤∀𝑧 𝑧 = 𝑤 → 𝑧 = 𝑥)
22	17, 21	syl 17	. . . . 5 ⊢ (∀𝑤(∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → 𝑧 = 𝑥)
23	13, 22	sylbi 217	. . . 4 ⊢ (∀𝑤∀𝑧(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → 𝑧 = 𝑥)
24	23	alcoms 2157	. . 3 ⊢ (∀𝑧∀𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → 𝑧 = 𝑥)
25	24	axc4i-o 38900	. 2 ⊢ (∀𝑧∀𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → ∀𝑧 𝑧 = 𝑥)
26	3, 10, 25	3syl 18	1 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376  ax-c5 38885  ax-c4 38886  ax-c7 38887  ax-c16 38894

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783

This theorem is referenced by: (None)