Theorem eu3 1436

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 1435	. 2 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
3	1	mo 1432	. . 3 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
4	3	anbi2i 483	. 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 990   = wceq 992  E.wex 1016  E!weu 1419

This theorem is referenced by:  mo2 1439  eu5 1448  2eu4 1492  reu6 1978  funeu 3642  eqeu 11394

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421

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