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	$$⊢\exists xx=y$$

Proof

Step	Hyp	Ref	Expression
1		19.2	$$⊢\forall xx=y\to \exists xx=y$$
2	1	a1d	$$⊢\forall xx=y\to \left(y=z\to \exists xx=y\right)$$
3		bj-ax6elem1	$$⊢\neg \forall xx=y\to \left(y=z\to \forall xy=z\right)$$
4		bj-ax6elem2	$$⊢\forall xy=z\to \exists xx=y$$
5	3 4	syl6	$$⊢\neg \forall xx=y\to \left(y=z\to \exists xx=y\right)$$
6	2 5	pm2.61i	$$⊢y=z\to \exists xx=y$$
7		ax6evr	$$⊢\exists zy=z$$
8	6 7	exlimiiv	$$⊢\exists xx=y$$