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Metamath Proof Explorer

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 )