Theorem sbie

Description: Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev and sbievw . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-1994) (Revised by Mario Carneiro, 4-Oct-2016) (Proof shortened by Wolf Lammen, 13-Jul-2019) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sbie.1	|- F/ x ps
2		sbie.2	|- ( x = y -> ( ph <-> ps ) )
3		equsb1	|- [ y / x ] x = y
4	2	sbimi	|- ( [ y / x ] x = y -> [ y / x ] ( ph <-> ps ) )
5	3 4	ax-mp	|- [ y / x ] ( ph <-> ps )
6	1	sbf	|- ( [ y / x ] ps <-> ps )
7	6	sblbis	|- ( [ y / x ] ( ph <-> ps ) <-> ( [ y / x ] ph <-> ps ) )
8	5 7	mpbi	|- ( [ y / x ] ph <-> ps )