Theorem cbvex4v 1918

Description: Rule used to change bound variables, using implicit substitition. (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, z, v   x, u, w, z   y, v, w, z

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 1856	. . 3 |- ( ( x = v /\ y = u ) -> ( E. z E. w ph <-> E. z E. w ps ) )
3	2	cbvex2v 1912	. 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 1912	. . 3 |- ( E. z E. w ps <-> E. f E. g ch )
6	5	2exbii 1594	. 2 |- ( E. v E. u E. z E. w ps <-> E. v E. u E. f E. g ch )
7	3, 6	bitri 183	1 |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch )

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:  enq0sym  7373  addnq0mo  7388  mulnq0mo  7389  addsrmo  7684  mulsrmo  7685