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

Metamath Proof Explorer

Theorem cbvex2vw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2vv with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

Ref Expression
Hypothesis cbval2vw.1 
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
Assertion cbvex2vw 
|- ( E. x E. y ph <-> E. z E. w ps )

Proof

Step Hyp Ref Expression
1 cbval2vw.1 
 |-  ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
2 1 cbvexdvaw 
 |-  ( x = z -> ( E. y ph <-> E. w ps ) )
3 2 cbvexvw 
 |-  ( E. x E. y ph <-> E. z E. w ps )