Theorem eujust 1773

Description: A soundness justification theorem for df-eu 1775, 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 1774 for a proof that provides an example of how it can be achieved through the use of dvelim 1743. (The proof was 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

Proof of Theorem eujust

Step	Hyp	Ref	Expression
1		equequ2 1495	. . . . 5 |- (y = w -> (x = y <-> x = w))
2	1	bibi2d 680	. . . 4 |- (y = w -> ((ph <-> x = y) <-> (ph <-> x = w)))
3	2	albidv 1656	. . 3 |- (y = w -> (A.x(ph <-> x = y) <-> A.x(ph <-> x = w)))
4	3	cbvexv 1697	. 2 |- (E.yA.x(ph <-> x = y) <-> E.wA.x(ph <-> x = w))
5		equequ2 1495	. . . . 5 |- (w = z -> (x = w <-> x = z))
6	5	bibi2d 680	. . . 4 |- (w = z -> ((ph <-> x = w) <-> (ph <-> x = z)))
7	6	albidv 1656	. . 3 |- (w = z -> (A.x(ph <-> x = w) <-> A.x(ph <-> x = z)))
8	7	cbvexv 1697	. 2 |- (E.wA.x(ph <-> x = w) <-> E.zA.x(ph <-> x = z))
9	4, 8	bitri 190	1 |- (E.yA.x(ph <-> x = y) <-> E.zA.x(ph <-> x = z))

Syntax hints:   <-> wb 163  A.wal 1296   = wceq 1298  E.wex 1326

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

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