Theorem cbvral3v 2846

Description: Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)

Hypotheses

Ref	Expression
cbvral3v.1	|- (x = w -> (ph <-> ch))
cbvral3v.2	|- (y = v -> (ch <-> th))
cbvral3v.3	|- (z = u -> (th <-> ps))

Assertion

Ref	Expression
cbvral3v	|- (A.x e. A A.y e. B A.z e. C ph <-> A.w e. A A.v e. B A.u e. C ps)

Distinct variable groups:   ph,w   ps,z   ch,x   ch,v   y,u,th   x,A   w,A   x,y,B   y,w,B   v,B   x,z,C,y   z,w,C   z,v,C   u,C

Allowed substitution hints:   ph(x,y,z,v,u)   ps(x,y,w,v,u)   ch(y,z,w,u)   th(x,z,w,v)   A(y,z,v,u)   B(z,u)

Proof of Theorem cbvral3v

Step	Hyp	Ref	Expression
1		cbvral3v.1	. . . 4 |- (x = w -> (ph <-> ch))
2	1	2ralbidv 2657	. . 3 |- (x = w -> (A.y e. B A.z e. C ph <-> A.y e. B A.z e. C ch))
3	2	cbvralv 2836	. 2 |- (A.x e. A A.y e. B A.z e. C ph <-> A.w e. A A.y e. B A.z e. C ch)
4		cbvral3v.2	. . . 4 |- (y = v -> (ch <-> th))
5		cbvral3v.3	. . . 4 |- (z = u -> (th <-> ps))
6	4, 5	cbvral2v 2844	. . 3 |- (A.y e. B A.z e. C ch <-> A.v e. B A.u e. C ps)
7	6	ralbii 2639	. 2 |- (A.w e. A A.y e. B A.z e. C ch <-> A.w e. A A.v e. B A.u e. C ps)
8	3, 7	bitri 240	1 |- (A.x e. A A.y e. B A.z e. C ph <-> A.w e. A A.v e. B A.u e. C ps)

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  A.wral 2615

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  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by: (None)