Theorem axc16g

Description: Generalization of axc16 . Use the latter when sufficient. This proof only requires, on top of { ax-1 -- ax-7 }, theorem ax12v . (Contributed by NM, 15-May-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 18-Feb-2018) Remove dependency on ax-13 , along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019) (Revised by BJ, 7-Jul-2021) Shorten axc11rv . (Revised by Wolf Lammen, 11-Oct-2021)

	Ref	Expression
Assertion	axc16g	$$⊢\forall xx=y\to \left(\phi \to \forall z\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		aevlem	$$⊢\forall xx=y\to \forall zz=w$$
2		ax12v	$$⊢z=w\to \left(\phi \to \forall z\left(z=w\to \phi \right)\right)$$
3	2	sps	$$⊢\forall zz=w\to \left(\phi \to \forall z\left(z=w\to \phi \right)\right)$$
4		pm2.27	$$⊢z=w\to \left(\left(z=w\to \phi \right)\to \phi \right)$$
5	4	al2imi	$$⊢\forall zz=w\to \left(\forall z\left(z=w\to \phi \right)\to \forall z\phi \right)$$
6	3 5	syld	$$⊢\forall zz=w\to \left(\phi \to \forall z\phi \right)$$
7	1 6	syl	$$⊢\forall xx=y\to \left(\phi \to \forall z\phi \right)$$