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	|- F/ x ps
	sbhypf.2	|- ( x = A -> ( ph <-> ps ) )
Assertion	sbhypf	|- ( y = A -> ( [ y / x ] ph <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		sbhypf.1	|- F/ x ps
2		sbhypf.2	|- ( x = A -> ( ph <-> ps ) )
3		eqeq1	|- ( x = y -> ( x = A <-> y = A ) )
4	3	equsexvw	|- ( E. x ( x = y /\ x = A ) <-> y = A )
5		nfs1v	|- F/ x [ y / x ] ph
6	5 1	nfbi	|- F/ x ( [ y / x ] ph <-> ps )
7		sbequ12	|- ( x = y -> ( ph <-> [ y / x ] ph ) )
8	7	bicomd	|- ( x = y -> ( [ y / x ] ph <-> ph ) )
9	8 2	sylan9bb	|- ( ( x = y /\ x = A ) -> ( [ y / x ] ph <-> ps ) )
10	6 9	exlimi	|- ( E. x ( x = y /\ x = A ) -> ( [ y / x ] ph <-> ps ) )
11	4 10	sylbir	|- ( y = A -> ( [ y / x ] ph <-> ps ) )