Theorem bj-axc10v

Description: Version of axc10 with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 14-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-axc10v	$$⊢\forall x\left(x=y\to \forall x\phi \right)\to \phi$$

Proof

Step	Hyp	Ref	Expression
1		ax6v	$$⊢\neg \forall x\neg x=y$$
2		con3	$$⊢\left(x=y\to \forall x\phi \right)\to \left(\neg \forall x\phi \to \neg x=y\right)$$
3	2	al2imi	$$⊢\forall x\left(x=y\to \forall x\phi \right)\to \left(\forall x\neg \forall x\phi \to \forall x\neg x=y\right)$$
4	1 3	mtoi	$$⊢\forall x\left(x=y\to \forall x\phi \right)\to \neg \forall x\neg \forall x\phi$$
5		axc7	$$⊢\neg \forall x\neg \forall x\phi \to \phi$$
6	4 5	syl	$$⊢\forall x\left(x=y\to \forall x\phi \right)\to \phi$$