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Metamath Proof Explorer

Theorem ralab2

Description: Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015) Drop ax-8 . (Revised by GG, 1-Dec-2023)

Ref Expression
Hypothesis ralab2.1 
|- ( x = y -> ( ps <-> ch ) )
Assertion ralab2 
|- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) )

Proof

Step Hyp Ref Expression
1 ralab2.1 
 |-  ( x = y -> ( ps <-> ch ) )
2 df-ral 
 |-  ( A. x e. { y | ph } ps <-> A. x ( x e. { y | ph } -> ps ) )
3 nfsab1 
 |-  F/ y x e. { y | ph }
4 nfv 
 |-  F/ y ps
5 3 4 nfim 
 |-  F/ y ( x e. { y | ph } -> ps )
6 nfv 
 |-  F/ x ( ph -> ch )
7 eleq1ab 
 |-  ( x = y -> ( x e. { y | ph } <-> y e. { y | ph } ) )
8 abid 
 |-  ( y e. { y | ph } <-> ph )
9 7 8 bitrdi 
 |-  ( x = y -> ( x e. { y | ph } <-> ph ) )
10 9 1 imbi12d 
 |-  ( x = y -> ( ( x e. { y | ph } -> ps ) <-> ( ph -> ch ) ) )
11 5 6 10 cbvalv1 
 |-  ( A. x ( x e. { y | ph } -> ps ) <-> A. y ( ph -> ch ) )
12 2 11 bitri 
 |-  ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) )