Theorem sbiev

Description: Conversion of implicit substitution to explicit substitution. Version of sbie with a disjoint variable condition, not requiring ax-13 . See sbievw for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by Wolf Lammen, 18-Jan-2023) Remove dependence on ax-10 and shorten proof. (Revised by BJ, 18-Jul-2023)

	Ref	Expression
Hypotheses	sbiev.1	|- F/ x ps
	sbiev.2	|- ( x = y -> ( ph <-> ps ) )
Assertion	sbiev	|- ( [ y / x ] ph <-> ps )

Proof

Step	Hyp	Ref	Expression
1		sbiev.1	|- F/ x ps
2		sbiev.2	|- ( x = y -> ( ph <-> ps ) )
3		sb6	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
4	1 2	equsalv	|- ( A. x ( x = y -> ph ) <-> ps )
5	3 4	bitri	|- ( [ y / x ] ph <-> ps )