Theorem ax13lem2 2348

Description: Lemma for nfeqf2 2349. This lemma is equivalent to ax13v 2345 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 2346	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
2		equeucl 2008	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑤 = 𝑦 → 𝑧 = 𝑤))
3	2	eximi 1816	. . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∃𝑥(𝑤 = 𝑦 → 𝑧 = 𝑤))
4		19.36v 1971	. . . . 5 ⊢ (∃𝑥(𝑤 = 𝑦 → 𝑧 = 𝑤) ↔ (∀𝑥 𝑤 = 𝑦 → 𝑧 = 𝑤))
5	3, 4	sylib 219	. . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∀𝑥 𝑤 = 𝑦 → 𝑧 = 𝑤))
6	1, 5	syl9 77	. . 3 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → (𝑤 = 𝑦 → 𝑧 = 𝑤)))
7	6	alrimdv 1907	. 2 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑤(𝑤 = 𝑦 → 𝑧 = 𝑤)))
8		equequ2 2010	. . 3 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦))
9	8	equsalvw 1987	. 2 ⊢ (∀𝑤(𝑤 = 𝑦 → 𝑧 = 𝑤) ↔ 𝑧 = 𝑦)
10	7, 9	syl6ib 252	1 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1520  ∃wex 1761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-13 2344

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762

This theorem is referenced by:  nfeqf2  2349  nfeqf2OLD  2350  wl-speqv  34314  wl-19.2reqv  34316