Theorem cbvex4v 2137

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

Hypotheses

Ref	Expression
cbvex4v.1	|- ( ( x = v /\ y = u ) -> ( ph <-> ps ) )
cbvex4v.2	|- ( ( z = f /\ w = g ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
cbvex4v	|- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch )

Distinct variable groups:   z, w, ch   v, u, ph   x, y, ps   f, g, ps   w, f   z, g   w, u, x, y, z, v

Allowed substitution hints:   ph( x, y, z, w, f, g)   ps( z, w, v, u)   ch( x, y, v, u, f, g)

Proof of Theorem cbvex4v

Step	Hyp	Ref	Expression
1		cbvex4v.1	. . . 4 |- ( ( x = v /\ y = u ) -> ( ph <-> ps ) )
2	1	2exbidv 1781	. . 3 |- ( ( x = v /\ y = u ) -> ( E. z E. w ph <-> E. z E. w ps ) )
3	2	cbvex2v 2134	. 2 |- ( E. x E. y E. z E. w ph <-> E. v E. u E. z E. w ps )
4		cbvex4v.2	. . . 4 |- ( ( z = f /\ w = g ) -> ( ps <-> ch ) )
5	4	cbvex2v 2134	. . 3 |- ( E. z E. w ps <-> E. f E. g ch )
6	5	2exbii 1730	. 2 |- ( E. v E. u E. z E. w ps <-> E. v E. u E. f E. g ch )
7	3, 6	bitri 257	1 |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   E.wex 1674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679

This theorem is referenced by:  addsrmo  9523  mulsrmo  9524