Theorem eujust 2472

Description: A soundness justification theorem for df-eu 2474, showing that the definition is equivalent to itself with its dummy variable renamed. Note that y and z needn't be distinct variables. See eujustALT 2473 for a proof that provides an example of how it can be achieved through the use of dvelim 2337. (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 1953	. . . . 5 |- ( y = w -> ( x = y <-> x = w ) )
2	1	bibi2d 332	. . . 4 |- ( y = w -> ( ( ph <-> x = y ) <-> ( ph <-> x = w ) ) )
3	2	albidv 1849	. . 3 |- ( y = w -> ( A. x ( ph <-> x = y ) <-> A. x ( ph <-> x = w ) ) )
4	3	cbvexv 2275	. 2 |- ( E. y A. x ( ph <-> x = y ) <-> E. w A. x ( ph <-> x = w ) )
5		equequ2 1953	. . . . 5 |- ( w = z -> ( x = w <-> x = z ) )
6	5	bibi2d 332	. . . 4 |- ( w = z -> ( ( ph <-> x = w ) <-> ( ph <-> x = z ) ) )
7	6	albidv 1849	. . 3 |- ( w = z -> ( A. x ( ph <-> x = w ) <-> A. x ( ph <-> x = z ) ) )
8	7	cbvexv 2275	. 2 |- ( E. w A. x ( ph <-> x = w ) <-> E. z A. x ( ph <-> x = z ) )
9	4, 8	bitri 264	1 |- ( E. y A. x ( ph <-> x = y ) <-> E. z A. x ( ph <-> x = z ) )

Syntax hints:   <-> wb 196   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)