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

Metamath Proof Explorer

Theorem cbv3h

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbv3hv if possible. (Contributed by NM, 8-Jun-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 12-May-2018) (New usage is discouraged.)

Ref Expression
Hypotheses cbv3h.1 
|- ( ph -> A. y ph )
cbv3h.2 
|- ( ps -> A. x ps )
cbv3h.3 
|- ( x = y -> ( ph -> ps ) )
Assertion cbv3h 
|- ( A. x ph -> A. y ps )

Proof

Step Hyp Ref Expression
1 cbv3h.1 
 |-  ( ph -> A. y ph )
2 cbv3h.2 
 |-  ( ps -> A. x ps )
3 cbv3h.3 
 |-  ( x = y -> ( ph -> ps ) )
4 1 nf5i 
 |-  F/ y ph
5 2 nf5i 
 |-  F/ x ps
6 4 5 3 cbv3 
 |-  ( A. x ph -> A. y ps )