Theorem sbhypf

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004) (Proof shortened by Wolf Lammen, 25-Jan-2025)

	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	2	sbimi	|- ( [ y / x ] x = A -> [ y / x ] ( ph <-> ps ) )
4		eqsb1	|- ( [ y / x ] x = A <-> y = A )
5	1	sbf	|- ( [ y / x ] ps <-> ps )
6	5	sblbis	|- ( [ y / x ] ( ph <-> ps ) <-> ( [ y / x ] ph <-> ps ) )
7	3 4 6	3imtr3i	|- ( y = A -> ( [ y / x ] ph <-> ps ) )