Theorem wl-ax11-lem2

Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019)

	Ref	Expression
Assertion	wl-ax11-lem2	$$⊢\left(\forall uu=y\wedge \neg \forall xx=y\right)\to \forall xu=y$$

Proof

Step	Hyp	Ref	Expression
1		sp	$$⊢\forall uu=y\to u=y$$
2		aev	$$⊢\forall xx=u\to \forall xx=y$$
3		pm2.21	$$⊢\neg \forall xx=y\to \left(\forall xx=y\to \forall xx=u\right)$$
4	2 3	impbid2	$$⊢\neg \forall xx=y\to \left(\forall xx=u\leftrightarrow \forall xx=y\right)$$
5	1 4	anim12i	$$⊢\left(\forall uu=y\wedge \neg \forall xx=y\right)\to \left(u=y\wedge \left(\forall xx=u\leftrightarrow \forall xx=y\right)\right)$$
6		wl-aleq	$$⊢\forall xu=y\leftrightarrow \left(u=y\wedge \left(\forall xx=u\leftrightarrow \forall xx=y\right)\right)$$
7	5 6	sylibr	$$⊢\left(\forall uu=y\wedge \neg \forall xx=y\right)\to \forall xu=y$$