Theorem cbvral3v 2667

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 2459	. . 3 |- ( x = w -> ( A. y e. B A. z e. C ph <-> A. y e. B A. z e. C ch ) )
3	2	cbvralv 2654	. 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 2665	. . 3 |- ( A. y e. B A. z e. C ch <-> A. v e. B A. u e. C ps )
7	6	ralbii 2441	. 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 183	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 104   A.wral 2416

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421

This theorem is referenced by: (None)