Theorem euf 1386

Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition.

Hypothesis

Ref	Expression
euf.1	|- (ph -> A.yph)

Assertion

Ref	Expression
euf	|- (E!xph <-> E.yA.x(ph <-> x = y))

Distinct variable group:   x,y

Proof of Theorem euf

Step	Hyp	Ref	Expression
1		df-eu 1384	. 2 |- (E!xph <-> E.zA.x(ph <-> x = z))
2		euf.1	. . . . 5 |- (ph -> A.yph)
3		ax-17 973	. . . . 5 |- (x = z -> A.y x = z)
4	2, 3	hbbi 1012	. . . 4 |- ((ph <-> x = z) -> A.y(ph <-> x = z))
5	4	hbal 1007	. . 3 |- (A.x(ph <-> x = z) -> A.yA.x(ph <-> x = z))
6		ax-17 973	. . 3 |- (A.x(ph <-> x = y) -> A.zA.x(ph <-> x = y))
7		equequ2 1137	. . . . 5 |- (z = y -> (x = z <-> x = y))
8	7	bibi2d 620	. . . 4 |- (z = y -> ((ph <-> x = z) <-> (ph <-> x = y)))
9	8	albidv 1280	. . 3 |- (z = y -> (A.x(ph <-> x = z) <-> A.x(ph <-> x = y)))
10	5, 6, 9	cbvex 1168	. 2 |- (E.zA.x(ph <-> x = z) <-> E.yA.x(ph <-> x = y))
11	1, 10	bitr 173	1 |- (E!xph <-> E.yA.x(ph <-> x = y))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956   = wceq 958  E.wex 982  E!weu 1382

This theorem is referenced by:  eu1 1394  eumo0 1397

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-eu 1384

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