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

Metamath Proof Explorer

Theorem cbvex2vv

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 cbvex2vw if possible. (Contributed by NM, 26-Jul-1995) Remove dependency on ax-10 . (Revised by Wolf Lammen, 18-Jul-2021) (New usage is discouraged.)

Ref Expression
Hypothesis cbval2vv.1 
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
Assertion cbvex2vv 
|- ( E. x E. y ph <-> E. z E. w ps )

Proof

Step Hyp Ref Expression
1 cbval2vv.1 
 |-  ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
2 1 cbvexdva 
 |-  ( x = z -> ( E. y ph <-> E. w ps ) )
3 2 cbvexv 
 |-  ( E. x E. y ph <-> E. z E. w ps )