Theorem sbiedv

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie ). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sbiedvw when possible. (Contributed by NM, 7-Jan-2017) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sbiedv.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	sbiedv	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbiedv.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		nfv	$$⊢Ⅎx\phi$$
3		nfvd	$$⊢\phi \to Ⅎx\chi$$
4	1	ex	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
5	2 3 4	sbied	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \chi \right)$$