Theorem cbvex2v 2287

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 18-Jul-2021.)

Hypothesis

Ref	Expression
cbval2v.1	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvex2v	|- ( E. x E. y ph <-> E. z E. w ps )

Distinct variable groups:   z, w, ph   x, y, ps   x, w   y, z

Allowed substitution hints:   ph( x, y)   ps( z, w)

Proof of Theorem cbvex2v

Step	Hyp	Ref	Expression
1		cbval2v.1	. . 3 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
2	1	cbvexdva 2283	. 2 |- ( x = z -> ( E. y ph <-> E. w ps ) )
3	2	cbvexv 2275	1 |- ( E. x E. y ph <-> E. z E. w ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  cbvex4v  2289  funop1  41302  uspgrsprf1  41755