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Theorem 2euex 1484
Description: Double quantification with existential uniqueness.
Assertion
Ref Expression
2euex |- (E!xE.yph -> E.yE!xph)

Proof of Theorem 2euex
StepHypRef Expression
1 eu5 1451 . 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
2 hbe1 1057 . . . . . . 7 |- (E.yph -> A.yE.yph)
32hbmo 1449 . . . . . 6 |- (E*xE.yph -> A.yE*xE.yph)
4319.41 1136 . . . . 5 |- (E.y(E.xph /\ E*xE.yph) <-> (E.yE.xph /\ E*xE.yph))
54biimpri 159 . . . 4 |- ((E.yE.xph /\ E*xE.yph) -> E.y(E.xph /\ E*xE.yph))
6 excom 1087 . . . 4 |- (E.xE.yph <-> E.yE.xph)
75, 6sylanb 460 . . 3 |- ((E.xE.yph /\ E*xE.yph) -> E.y(E.xph /\ E*xE.yph))
8 2moex 1483 . . . . . . 7 |- (E*xE.yph -> A.yE*xph)
9819.21bi 1101 . . . . . 6 |- (E*xE.yph -> E*xph)
109anim2i 342 . . . . 5 |- ((E.xph /\ E*xE.yph) -> (E.xph /\ E*xph))
11 eu5 1451 . . . . 5 |- (E!xph <-> (E.xph /\ E*xph))
1210, 11sylibr 207 . . . 4 |- ((E.xph /\ E*xE.yph) -> E!xph)
131219.22i 1081 . . 3 |- (E.y(E.xph /\ E*xE.yph) -> E.yE!xph)
147, 13syl 10 . 2 |- ((E.xE.yph /\ E*xE.yph) -> E.yE!xph)
151, 14sylbi 206 1 |- (E!xE.yph -> E.yE!xph)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 230  E.wex 1021  E!weu 1422  E*wmo 1423
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260
This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425
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