Theorem cbvex2v 1912

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)

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		nfv 1516	. 2 |- F/ z ph
2		nfv 1516	. 2 |- F/ w ph
3		nfv 1516	. 2 |- F/ x ps
4		nfv 1516	. 2 |- F/ y ps
5		cbval2v.1	. 2 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvex2 1910	1 |- ( E. x E. y ph <-> E. z E. w ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  cbvex4v  1918  th3qlem1  6603