Theorem bj-ax6e 37347

Description: Proof of ax6e 2417 (hence ax6 2418) from Tarski's system, ax-c9 39722, ax-c16 39724. Remark: ax-6 2000 is used only via its principal (unbundled) instance ax6v 2001. (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 2009	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2	1	a1d 26	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
3		bj-ax6elem1 37345	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
4		bj-ax6elem2 37346	. . . 4 ⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
5	3, 4	syl6 36	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
6	2, 5	pm2.61i 184	. 2 ⊢ (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
7		ax6evr 2048	. 2 ⊢ ∃𝑧 𝑦 = 𝑧
8	6, 7	exlimiiv 1964	1 ⊢ ∃𝑥 𝑥 = 𝑦

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by: (None)