Theorem bj-ax6e

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

	Ref	Expression
Assertion	bj-ax6e	⊢ ∃ 𝑥 𝑥 = 𝑦

Proof

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