Theorem eualeq 3823

Description: Two ways to express single-valuedness of a class expression A(x).

Assertion

Ref	Expression
eualeq	|- (E!yA.x y = A <-> E.yA.x y = A)

Distinct variable groups:   x,y   y,A

Proof of Theorem eualeq

Step	Hyp	Ref	Expression
1		eqeq1 1890	. . . 4 |- (y = z -> (y = A <-> z = A))
2	1	albidv 1656	. . 3 |- (y = z -> (A.x y = A <-> A.x z = A))
3	2	eu4 1806	. 2 |- (E!yA.x y = A <-> (E.yA.x y = A /\ A.yA.z((A.x y = A /\ A.x z = A) -> y = z)))
4		eqtr3 1907	. . . 4 |- ((y = A /\ z = A) -> y = z)
5		ax-4 1319	. . . 4 |- (A.x y = A -> y = A)
6		ax-4 1319	. . . 4 |- (A.x z = A -> z = A)
7	4, 5, 6	syl2an 503	. . 3 |- ((A.x y = A /\ A.x z = A) -> y = z)
8	7	gen2 1329	. 2 |- A.yA.z((A.x y = A /\ A.x z = A) -> y = z)
9	3, 8	mpbiran2 799	1 |- (E!yA.x y = A <-> E.yA.x y = A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326  E!weu 1771

This theorem is referenced by:  euexaleq 3827

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-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-cleq 1877

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