Theorem bj-subst

Description: Proof of sbalex from core axioms, ax-10 (modal5), and bj-ax12 . (Contributed by BJ, 29-Dec-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-subst	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-ax12	$$⊢\forall x\left(x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$
2		pm3.31	$$⊢\left(x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)\to \left(\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \phi \right)\right)$$
3	2	aleximi	$$⊢\forall x\left(x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)\to \left(\exists x\left(x=y\wedge \phi \right)\to \exists x\forall x\left(x=y\to \phi \right)\right)$$
4	1 3	ax-mp	$$⊢\exists x\left(x=y\wedge \phi \right)\to \exists x\forall x\left(x=y\to \phi \right)$$
5		hbe1a	$$⊢\exists x\forall x\left(x=y\to \phi \right)\to \forall x\left(x=y\to \phi \right)$$
6	4 5	syl	$$⊢\exists x\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \phi \right)$$
7		equs4v	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$
8	6 7	impbii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)$$