Theorem sbie

Description: Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev and sbievw . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-1994) (Revised by Mario Carneiro, 4-Oct-2016) (Proof shortened by Wolf Lammen, 13-Jul-2019) (New usage is discouraged.)

	Ref	Expression
Hypotheses	sbie.1	$$⊢Ⅎx\psi$$
	sbie.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	sbie	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		sbie.1	$$⊢Ⅎx\psi$$
2		sbie.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3		equsb1	$$⊢\left[y/x\right]x=y$$
4	2	sbimi	$$⊢\left[y/x\right]x=y\to \left[y/x\right]\left(\phi \leftrightarrow \psi \right)$$
5	3 4	ax-mp	$$⊢\left[y/x\right]\left(\phi \leftrightarrow \psi \right)$$
6	1	sbf	$$⊢\left[y/x\right]\psi \leftrightarrow \psi$$
7	6	sblbis	$$⊢\left[y/x\right]\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \leftrightarrow \psi \right)$$
8	5 7	mpbi	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$