Metamath Proof Explorer

Theorem abbibOLD

Description: Obsolete version of abbib as of 22-Oct-2026. (Contributed by NM, 25-Nov-2013) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion abbibOLD 
|- ( { x | ph } = { x | ps } <-> A. x ( ph <-> ps ) )

Proof

Step Hyp Ref Expression
1 dfcleq 
 |-  ( { x | ph } = { x | ps } <-> A. y ( y e. { x | ph } <-> y e. { x | ps } ) )
2 nfsab1 
 |-  F/ x y e. { x | ph }
3 nfsab1 
 |-  F/ x y e. { x | ps }
4 2 3 nfbi 
 |-  F/ x ( y e. { x | ph } <-> y e. { x | ps } )
5 nfv 
 |-  F/ y ( ph <-> ps )
6 df-clab 
 |-  ( y e. { x | ph } <-> [ y / x ] ph )
7 sbequ12r 
 |-  ( y = x -> ( [ y / x ] ph <-> ph ) )
8 6 7 bitrid 
 |-  ( y = x -> ( y e. { x | ph } <-> ph ) )
9 df-clab 
 |-  ( y e. { x | ps } <-> [ y / x ] ps )
10 sbequ12r 
 |-  ( y = x -> ( [ y / x ] ps <-> ps ) )
11 9 10 bitrid 
 |-  ( y = x -> ( y e. { x | ps } <-> ps ) )
12 8 11 bibi12d 
 |-  ( y = x -> ( ( y e. { x | ph } <-> y e. { x | ps } ) <-> ( ph <-> ps ) ) )
13 4 5 12 cbvalv1 
 |-  ( A. y ( y e. { x | ph } <-> y e. { x | ps } ) <-> A. x ( ph <-> ps ) )
14 1 13 bitri 
 |-  ( { x | ph } = { x | ps } <-> A. x ( ph <-> ps ) )