Theorem eu3 1390

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 1389	. 2 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
3	1	mo 1386	. . 3 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
4	3	anbi2i 479	. 2 |- ((E.xph /\ E.yA.x(ph -> x = y)) <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
5	2, 4	bitr4 176	1 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  E!weu 1373

This theorem is referenced by:  mo2 1393  eu5 1402  2eu4 1445  reu6 1922  funeu 3523

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375

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