Theorem wl-ax13lem1 33287

Description: A version of ax-wl-13v 33286 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.)

Assertion

Ref	Expression
wl-ax13lem1	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Distinct variable group:   𝑥,𝑧

Proof of Theorem wl-ax13lem1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equviniva 1960	. 2 ⊢ (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤 ∧ 𝑦 = 𝑤))
2		ax-wl-13v 33286	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑤 → ∀𝑥 𝑦 = 𝑤))
3		equeucl 1951	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑦 = 𝑤 → 𝑧 = 𝑦))
4	3	alimdv 1845	. . . . 5 ⊢ (𝑧 = 𝑤 → (∀𝑥 𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦))
5	2, 4	syl9 77	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑤 → (𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦)))
6	5	impd 447	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑧 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
7	6	exlimdv 1861	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑤(𝑧 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
8	1, 7	syl5 34	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-wl-13v 33286

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)