Theorem eu3 1397

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 1396	. 2 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
3	1	mo 1393	. . 3 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
4	3	anbi2i 480	. 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 954   = wceq 956  E.wex 980  E!weu 1380

This theorem is referenced by:  mo2 1400  eu5 1409  2eu4 1452  reu6 1932  funeu 3537

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382

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