Theorem eujust 2047

Description: A soundness justification theorem for df-eu 2048, showing that the definition is equivalent to itself with its dummy variable renamed. Note that y and z needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
eujust	|- ( E. y A. x ( ph <-> x = y ) <-> E. z A. x ( ph <-> x = z ) )

Distinct variable groups:   x, y   x, z   ph, y   ph, z

Allowed substitution hint:   ph( x)

Proof of Theorem eujust

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1727	. . . . 5 |- ( y = w -> ( x = y <-> x = w ) )
2	1	bibi2d 232	. . . 4 |- ( y = w -> ( ( ph <-> x = y ) <-> ( ph <-> x = w ) ) )
3	2	albidv 1838	. . 3 |- ( y = w -> ( A. x ( ph <-> x = y ) <-> A. x ( ph <-> x = w ) ) )
4	3	cbvexv 1933	. 2 |- ( E. y A. x ( ph <-> x = y ) <-> E. w A. x ( ph <-> x = w ) )
5		equequ2 1727	. . . . 5 |- ( w = z -> ( x = w <-> x = z ) )
6	5	bibi2d 232	. . . 4 |- ( w = z -> ( ( ph <-> x = w ) <-> ( ph <-> x = z ) ) )
7	6	albidv 1838	. . 3 |- ( w = z -> ( A. x ( ph <-> x = w ) <-> A. x ( ph <-> x = z ) ) )
8	7	cbvexv 1933	. 2 |- ( E. w A. x ( ph <-> x = w ) <-> E. z A. x ( ph <-> x = z ) )
9	4, 8	bitri 184	1 |- ( E. y A. x ( ph <-> x = y ) <-> E. z A. x ( ph <-> x = z ) )

Syntax hints:   <-> wb 105   A.wal 1362   = wceq 1364   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)