Theorem bj-axc10

Description: Alternate proof of axc10 . Shorter. One can prove a version with DV ( x , y ) without ax-13 , by using ax6ev instead of ax6e . (Contributed by BJ, 31-Mar-2021) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-axc10	$$⊢\forall x\left(x=y\to \forall x\phi \right)\to \phi$$

Proof

Step	Hyp	Ref	Expression
1		ax6e	$$⊢\exists xx=y$$
2		exim	$$⊢\forall x\left(x=y\to \forall x\phi \right)\to \left(\exists xx=y\to \exists x\forall x\phi \right)$$
3	1 2	mpi	$$⊢\forall x\left(x=y\to \forall x\phi \right)\to \exists x\forall x\phi$$
4		axc7e	$$⊢\exists x\forall x\phi \to \phi$$
5	3 4	syl	$$⊢\forall x\left(x=y\to \forall x\phi \right)\to \phi$$