Theorem wl-19.2reqv

Description: Under the assumption -. x = y the reverse direction of 19.2 is provable from Tarski's FOL and ax13v only. Note that in conjunction with 19.2 in fact ( -. x = y -> ( A. x z = y <-> E. x z = y ) ) holds. (Contributed by Wolf Lammen, 17-Apr-2021)

	Ref	Expression
Assertion	wl-19.2reqv	$$⊢\neg x=y\to \left(\exists xz=y\to \forall xz=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax13lem2	$$⊢\neg x=y\to \left(\exists xz=y\to z=y\right)$$
2		ax13lem1	$$⊢\neg x=y\to \left(z=y\to \forall xz=y\right)$$
3	1 2	syld	$$⊢\neg x=y\to \left(\exists xz=y\to \forall xz=y\right)$$