Theorem cbvex4v 2012

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.xE.yE.zE.wph <-> E.vE.uE.fE.gch)

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 1628	. . 3 |- ((x = v /\ y = u) -> (E.zE.wph <-> E.zE.wps))
3	2	cbvex2v 2007	. 2 |- (E.xE.yE.zE.wph <-> E.vE.uE.zE.wps)
4		cbvex4v.2	. . . 4 |- ((z = f /\ w = g) -> (ps <-> ch))
5	4	cbvex2v 2007	. . 3 |- (E.zE.wps <-> E.fE.gch)
6	5	2exbii 1583	. 2 |- (E.vE.uE.zE.wps <-> E.vE.uE.fE.gch)
7	3, 6	bitri 240	1 |- (E.xE.yE.zE.wph <-> E.vE.uE.fE.gch)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)