Theorem bj-ax6e 34114

Description: Proof of ax6e 2390 (hence ax6 2391) from Tarski's system, ax-c9 36186, ax-c16 36188. Remark: ax-6 1970 is used only via its principal (unbundled) instance ax6v 1971. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-ax6e	⊢ ∃𝑥 𝑥 = 𝑦

Proof of Theorem bj-ax6e

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.2 1981	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2	1	a1d 25	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
3		bj-ax6elem1 34112	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
4		bj-ax6elem2 34113	. . . 4 ⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
5	3, 4	syl6 35	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
6	2, 5	pm2.61i 185	. 2 ⊢ (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
7		ax6evr 2022	. 2 ⊢ ∃𝑧 𝑦 = 𝑧
8	6, 7	exlimiiv 1932	1 ⊢ ∃𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)