Theorem 2ax6e

Description: We can always find values matching x and y , as long as they are represented by distinct variables. Version of 2ax6elem with a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Wolf Lammen, 28-Sep-2018) (Proof shortened by Wolf Lammen, 3-Oct-2023) (New usage is discouraged.)

	Ref	Expression
Assertion	2ax6e	$$⊢\exists z\exists w\left(z=x\wedge w=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		aeveq	$$⊢\forall ww=z\to z=x$$
2		aeveq	$$⊢\forall ww=z\to w=y$$
3	1 2	jca	$$⊢\forall ww=z\to \left(z=x\wedge w=y\right)$$
4	3	19.8ad	$$⊢\forall ww=z\to \exists w\left(z=x\wedge w=y\right)$$
5	4	19.8ad	$$⊢\forall ww=z\to \exists z\exists w\left(z=x\wedge w=y\right)$$
6		2ax6elem	$$⊢\neg \forall ww=z\to \exists z\exists w\left(z=x\wedge w=y\right)$$
7	5 6	pm2.61i	$$⊢\exists z\exists w\left(z=x\wedge w=y\right)$$