Theorem sbrimvlem

Description: Common proof template for sbrimvw and sbrimv . The hypothesis is an instance of 19.21 . (Contributed by Wolf Lammen, 29-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		sbrimvlem.1	|- ( A. x ( ph -> ( x = y -> ps ) ) <-> ( ph -> A. x ( x = y -> ps ) ) )
2		sb6	|- ( [ y / x ] ( ph -> ps ) <-> A. x ( x = y -> ( ph -> ps ) ) )
3		bi2.04	|- ( ( ph -> ( x = y -> ps ) ) <-> ( x = y -> ( ph -> ps ) ) )
4	3	albii	|- ( A. x ( ph -> ( x = y -> ps ) ) <-> A. x ( x = y -> ( ph -> ps ) ) )
5	2 4 1	3bitr2i	|- ( [ y / x ] ( ph -> ps ) <-> ( ph -> A. x ( x = y -> ps ) ) )
6		sb6	|- ( [ y / x ] ps <-> A. x ( x = y -> ps ) )
7	6	imbi2i	|- ( ( ph -> [ y / x ] ps ) <-> ( ph -> A. x ( x = y -> ps ) ) )
8	5 7	bitr4i	|- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )