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Theorem 2euex 2133
Description: Double quantification with existential uniqueness. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
2euex |- (E!xE.yph -> E.yE!xph)

Proof of Theorem 2euex
StepHypRef Expression
1 eu5 2099 . 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
2 excom 1711 . . . 4 |- (E.xE.yph <-> E.yE.xph)
3 hbe1 1681 . . . . . 6 |- (E.yph -> A.yE.yph)
43hbmo 2097 . . . . 5 |- (E*xE.yph -> A.yE*xE.yph)
5 19.8a 1694 . . . . . . 7 |- (ph -> E.yph)
65immoi 2108 . . . . . 6 |- (E*xE.yph -> E*xph)
7 df-mo 2071 . . . . . 6 |- (E*xph <-> (E.xph -> E!xph))
86, 7sylib 263 . . . . 5 |- (E*xE.yph -> (E.xph -> E!xph))
94, 8eximd 1727 . . . 4 |- (E*xE.yph -> (E.yE.xph -> E.yE!xph))
102, 9syl5bi 270 . . 3 |- (E*xE.yph -> (E.xE.yph -> E.yE!xph))
1110impcom 490 . 2 |- ((E.xE.yph /\ E*xE.yph) -> E.yE!xph)
121, 11sylbi 237 1 |- (E!xE.yph -> E.yE!xph)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 433  E.wex 1644  E!weu 2066  E*wmo 2067
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-9 1624  ax-10 1625  ax-11 1626  ax-12 1627  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893
This theorem depends on definitions:  df-bi 232  df-or 434  df-an 435  df-ex 1645  df-sb 1845  df-eu 2070  df-mo 2071
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