This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cbval

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out cbvalw , cbvalvw , cbvalv1 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993) (Revised by Mario Carneiro, 3-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses cbval.1 
|- F/ y ph
cbval.2 
|- F/ x ps
cbval.3 
|- ( x = y -> ( ph <-> ps ) )
Assertion cbval 
|- ( A. x ph <-> A. y ps )

Proof

Step Hyp Ref Expression
1 cbval.1 
 |-  F/ y ph
2 cbval.2 
 |-  F/ x ps
3 cbval.3 
 |-  ( x = y -> ( ph <-> ps ) )
4 3 biimpd 
 |-  ( x = y -> ( ph -> ps ) )
5 1 2 4 cbv3 
 |-  ( A. x ph -> A. y ps )
6 3 biimprd 
 |-  ( x = y -> ( ps -> ph ) )
7 6 equcoms 
 |-  ( y = x -> ( ps -> ph ) )
8 2 1 7 cbv3 
 |-  ( A. y ps -> A. x ph )
9 5 8 impbii 
 |-  ( A. x ph <-> A. y ps )