Theorem cbvral3v 2707

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 2490	. . 3 |- ( x = w -> ( A. y e. B A. z e. C ph <-> A. y e. B A. z e. C ch ) )
3	2	cbvralv 2692	. 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 2705	. . 3 |- ( A. y e. B A. z e. C ch <-> A. v e. B A. u e. C ps )
7	6	ralbii 2472	. 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 2444

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449

This theorem is referenced by: (None)