Theorem bj-ax6e 32653

Description: Proof of ax6e 2250 (hence ax6 2251) from Tarski's system, ax-c9 34175, ax-c16 34177. Remark: ax-6 1888 is used only via its principal (unbundled) instance ax6v 1889. (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 1892	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2	1	a1d 25	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
3		bj-ax6elem1 32651	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
4		bj-ax6elem2 32652	. . . 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 176	. 2 ⊢ (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
7		ax6evr 1942	. 2 ⊢ ∃𝑧 𝑦 = 𝑧
8	6, 7	exlimiiv 1859	1 ⊢ ∃𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∀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-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)