Theorem sbhypf

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004)

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

Proof

Step	Hyp	Ref	Expression
1		sbhypf.1	$$⊢Ⅎx\psi$$
2		sbhypf.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		eqeq1	$$⊢x=y\to \left(x=A\leftrightarrow y=A\right)$$
4	3	equsexvw	$$⊢\exists x\left(x=y\wedge x=A\right)\leftrightarrow y=A$$
5		nfs1v	$$⊢Ⅎx\left[y/x\right]\phi$$
6	5 1	nfbi	$$⊢Ⅎx\left(\left[y/x\right]\phi \leftrightarrow \psi \right)$$
7		sbequ12	$$⊢x=y\to \left(\phi \leftrightarrow \left[y/x\right]\phi \right)$$
8	7	bicomd	$$⊢x=y\to \left(\left[y/x\right]\phi \leftrightarrow \phi \right)$$
9	8 2	sylan9bb	$$⊢\left(x=y\wedge x=A\right)\to \left(\left[y/x\right]\phi \leftrightarrow \psi \right)$$
10	6 9	exlimi	$$⊢\exists x\left(x=y\wedge x=A\right)\to \left(\left[y/x\right]\phi \leftrightarrow \psi \right)$$
11	4 10	sylbir	$$⊢y=A\to \left(\left[y/x\right]\phi \leftrightarrow \psi \right)$$