Theorem ax12v2

Description: It is possible to remove any restriction on ph in ax12v . Same as Axiom C8 of Monk2 p. 105. Use ax12v instead when sufficient. (Contributed by NM, 5-Aug-1993) Remove dependencies on ax-10 and ax-13 . (Revised by Jim Kingdon, 15-Dec-2017) (Proof shortened by Wolf Lammen, 8-Dec-2019)

	Ref	Expression
Assertion	ax12v2	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		equtrr	$$⊢y=z\to \left(x=y\to x=z\right)$$
2		ax12v	$$⊢x=z\to \left(\phi \to \forall x\left(x=z\to \phi \right)\right)$$
3	1	imim1d	$$⊢y=z\to \left(\left(x=z\to \phi \right)\to \left(x=y\to \phi \right)\right)$$
4	3	alimdv	$$⊢y=z\to \left(\forall x\left(x=z\to \phi \right)\to \forall x\left(x=y\to \phi \right)\right)$$
5	2 4	syl9r	$$⊢y=z\to \left(x=z\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$
6	1 5	syld	$$⊢y=z\to \left(x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$
7		ax6evr	$$⊢\exists zy=z$$
8	6 7	exlimiiv	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$