Theorem eqeu 3007

Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)

Hypothesis

Ref	Expression
eqeu.1	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
eqeu	|- ((A e. B /\ ps /\ A.x(ph -> x = A)) -> E!xph)

Distinct variable groups:   ps,x   x,A

Allowed substitution hints:   ph(x)   B(x)

Proof of Theorem eqeu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeu.1	. . . . 5 |- (x = A -> (ph <-> ps))
2	1	spcegv 2940	. . . 4 |- (A e. B -> (ps -> E.xph))
3	2	imp 418	. . 3 |- ((A e. B /\ ps) -> E.xph)
4	3	3adant3 975	. 2 |- ((A e. B /\ ps /\ A.x(ph -> x = A)) -> E.xph)
5		eqeq2 2362	. . . . . . 7 |- (y = A -> (x = y <-> x = A))
6	5	imbi2d 307	. . . . . 6 |- (y = A -> ((ph -> x = y) <-> (ph -> x = A)))
7	6	albidv 1625	. . . . 5 |- (y = A -> (A.x(ph -> x = y) <-> A.x(ph -> x = A)))
8	7	spcegv 2940	. . . 4 |- (A e. B -> (A.x(ph -> x = A) -> E.yA.x(ph -> x = y)))
9	8	imp 418	. . 3 |- ((A e. B /\ A.x(ph -> x = A)) -> E.yA.x(ph -> x = y))
10	9	3adant2 974	. 2 |- ((A e. B /\ ps /\ A.x(ph -> x = A)) -> E.yA.x(ph -> x = y))
11		nfv 1619	. . 3 |- F/yph
12	11	eu3 2230	. 2 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))
13	4, 10, 12	sylanbrc 645	1 |- ((A e. B /\ ps /\ A.x(ph -> x = A)) -> E!xph)

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204

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  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by: (None)