Theorem cbvralsvw

Description: Change bound variable by using a substitution. Version of cbvralsv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 20-Nov-2005) Avoid ax-13 . (Revised by GG, 10-Jan-2024) (Proof shortened by Wolf Lammen, 8-Mar-2025) Avoid ax-10 , ax-12 . (Revised by SN, 21-Aug-2025)

	Ref	Expression
Assertion	cbvralsvw	|- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		sb8v	|- ( A. x ( x e. A -> ph ) <-> A. y [ y / x ] ( x e. A -> ph ) )
2		df-ral	|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3		df-ral	|- ( A. y e. A [ y / x ] ph <-> A. y ( y e. A -> [ y / x ] ph ) )
4		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
5	4	imbi1d	|- ( x = y -> ( ( x e. A -> ph ) <-> ( y e. A -> ph ) ) )
6	5	sbbiiev	|- ( [ y / x ] ( x e. A -> ph ) <-> [ y / x ] ( y e. A -> ph ) )
7		sbrimvw	|- ( [ y / x ] ( y e. A -> ph ) <-> ( y e. A -> [ y / x ] ph ) )
8	6 7	bitr2i	|- ( ( y e. A -> [ y / x ] ph ) <-> [ y / x ] ( x e. A -> ph ) )
9	8	albii	|- ( A. y ( y e. A -> [ y / x ] ph ) <-> A. y [ y / x ] ( x e. A -> ph ) )
10	3 9	bitri	|- ( A. y e. A [ y / x ] ph <-> A. y [ y / x ] ( x e. A -> ph ) )
11	1 2 10	3bitr4i	|- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )