Theorem ax13lem1 2406

Description: A version of ax13v 2405 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2407 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑧

Proof of Theorem ax13lem1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinva 2060	. 2 ⊢ (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤 ∧ 𝑦 = 𝑤))
2		ax13v 2405	. . . . 5 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑤 → ∀𝑥 𝑦 = 𝑤))
3		equeucl 2054	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑦 = 𝑤 → 𝑧 = 𝑦))
4	3	alimdv 1946	. . . . 5 ⊢ (𝑧 = 𝑤 → (∀𝑥 𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦))
5	2, 4	syl9 78	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑤 → (𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦)))
6	5	impd 415	. . 3 ⊢ (¬ 𝑥 = 𝑦 → ((𝑧 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
7	6	exlimdv 1963	. 2 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑤(𝑧 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
8	1, 7	syl5 35	1 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  ax13  2407  ax13lem2  2408  nfeqf2  2409  ax6e  2415  wl-19.8eqv  38159  wl-19.2reqv  38160