Theorem ax13lem2 2410

Description: Lemma for nfeqf2 2411. This lemma is equivalent to ax13v 2407 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
ax13lem2	⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))

Distinct variable group:   𝑥,𝑧

Proof of Theorem ax13lem2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax13lem1 2408	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
2		equeucl 2047	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑤 = 𝑦 → 𝑧 = 𝑤))
3	2	eximi 1858	. . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∃𝑥(𝑤 = 𝑦 → 𝑧 = 𝑤))
4		19.36v 2016	. . . . 5 ⊢ (∃𝑥(𝑤 = 𝑦 → 𝑧 = 𝑤) ↔ (∀𝑥 𝑤 = 𝑦 → 𝑧 = 𝑤))
5	3, 4	sylib 221	. . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∀𝑥 𝑤 = 𝑦 → 𝑧 = 𝑤))
6	1, 5	syl9 78	. . 3 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → (𝑤 = 𝑦 → 𝑧 = 𝑤)))
7	6	alrimdv 1952	. 2 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑤(𝑤 = 𝑦 → 𝑧 = 𝑤)))
8		equequ2 2049	. . 3 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦))
9	8	equsalvw 2027	. 2 ⊢ (∀𝑤(𝑤 = 𝑦 → 𝑧 = 𝑤) ↔ 𝑧 = 𝑦)
10	7, 9	imbitrdi 254	1 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-13 2406

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  nfeqf2  2411  wl-speqv  38037  wl-19.2reqv  38039