Theorem cbvex2v 1991

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 1674	. 2 |- F/ z ph
2		nfv 1674	. 2 |- F/ w ph
3		nfv 1674	. 2 |- F/ x ps
4		nfv 1674	. 2 |- F/ y ps
5		cbval2v.1	. 2 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvex2 1988	1 |- ( E. x E. y ph <-> E. z E. w ps )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   E.wex 1587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591

This theorem is referenced by:  cbvex4v  1994  2moOLDOLD  2371  2eu6OLD  2381  th3qlem1  7315