Theorem eu3 1430

Description: An alternate way to express existential uniqueness.

Hypothesis

Ref	Expression
eu3.1	|- (ph -> A.yph)

Assertion

Ref	Expression
eu3	|- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))

Distinct variable group:   x,y

Proof of Theorem eu3

Step	Hyp	Ref	Expression
1		eu3.1	. . 3 |- (ph -> A.yph)
2	1	eu2 1429	. 2 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
3	1	mo 1426	. . 3 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
4	3	anbi2i 482	. 2 |- ((E.xph /\ E.yA.x(ph -> x = y)) <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
5	2, 4	bitr4i 174	1 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 986   = wceq 988  E.wex 1012  E!weu 1413

This theorem is referenced by:  mo2 1433  eu5 1442  2eu4 1486  reu6 1970  funeu 3612

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415

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