Theorem eujust 2206

Description: A soundness justification theorem for df-eu 2208, 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 2207 for a proof that provides an example of how it can be achieved through the use of dvelim 2016. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
eujust	|- (E.yA.x(ph <-> x = y) <-> E.zA.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 1686	. . . . 5 |- (y = w -> (x = y <-> x = w))
2	1	bibi2d 309	. . . 4 |- (y = w -> ((ph <-> x = y) <-> (ph <-> x = w)))
3	2	albidv 1625	. . 3 |- (y = w -> (A.x(ph <-> x = y) <-> A.x(ph <-> x = w)))
4	3	cbvexv 2003	. 2 |- (E.yA.x(ph <-> x = y) <-> E.wA.x(ph <-> x = w))
5		equequ2 1686	. . . . 5 |- (w = z -> (x = w <-> x = z))
6	5	bibi2d 309	. . . 4 |- (w = z -> ((ph <-> x = w) <-> (ph <-> x = z)))
7	6	albidv 1625	. . 3 |- (w = z -> (A.x(ph <-> x = w) <-> A.x(ph <-> x = z)))
8	7	cbvexv 2003	. 2 |- (E.wA.x(ph <-> x = w) <-> E.zA.x(ph <-> x = z))
9	4, 8	bitri 240	1 |- (E.yA.x(ph <-> x = y) <-> E.zA.x(ph <-> x = z))

Syntax hints:   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)