Theorem bj-axc14

Description: Alternate proof of axc14 (even when inlining the above results, this gives a shorter proof). (Contributed by BJ, 20-Oct-2021) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-axc14	$$⊢\neg \forall zz=x\to \left(\neg \forall zz=y\to \left(x\in y\to \forall zx\in y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-axc14nf	$$⊢\neg \forall zz=x\to \left(\neg \forall zz=y\to Ⅎzx\in y\right)$$
2		nf5r	$$⊢Ⅎzx\in y\to \left(x\in y\to \forall zx\in y\right)$$
3	2	a1i	$$⊢\neg \forall zz=x\to \left(Ⅎzx\in y\to \left(x\in y\to \forall zx\in y\right)\right)$$
4	1 3	syld	$$⊢\neg \forall zz=x\to \left(\neg \forall zz=y\to \left(x\in y\to \forall zx\in y\right)\right)$$