Theorem bj-ax12v3

Description: A weak version of ax-12 which is stronger than ax12v . Note that if one assumes reflexivity of equality |- x = x ( equid ), then bj-ax12v3 implies ax-5 over modal logic K (substitute x for y ). See also bj-ax12v3ALT . (Contributed by BJ, 6-Jul-2021) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ax12v3	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-5	$$⊢\phi \to \forall y\phi$$
2		ax12	$$⊢x=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)$$
3	1 2	syl5	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$