Theorem sbcieg

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 12-Oct-2024)

	Ref	Expression
Hypothesis	sbcieg.1	|- ( x = A -> ( ph <-> ps ) )
Assertion	sbcieg	|- ( A e. V -> ( [. A / x ]. ph <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		sbcieg.1	|- ( x = A -> ( ph <-> ps ) )
2		df-sbc	|- ( [. A / x ]. ph <-> A e. { x | ph } )
3	1	elabg	|- ( A e. V -> ( A e. { x | ph } <-> ps ) )
4	2 3	bitrid	|- ( A e. V -> ( [. A / x ]. ph <-> ps ) )