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Theorem 2euswap 1488
Description: A condition allowing swap of uniqueness and existential quantifiers.
Assertion
Ref Expression
2euswap |- (A.xE*yph -> (E!xE.yph -> E!yE.xph))

Proof of Theorem 2euswap
StepHypRef Expression
1 excomim 1086 . . . 4 |- (E.xE.yph -> E.yE.xph)
21a1i 8 . . 3 |- (A.xE*yph -> (E.xE.yph -> E.yE.xph))
3 2moswap 1487 . . 3 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
42, 3anim12d 569 . 2 |- (A.xE*yph -> ((E.xE.yph /\ E*xE.yph) -> (E.yE.xph /\ E*yE.xph)))
5 eu5 1451 . 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
6 eu5 1451 . 2 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
74, 5, 63imtr4g 564 1 |- (A.xE*yph -> (E!xE.yph -> E!yE.xph))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 230  A.wal 995  E.wex 1021  E!weu 1422  E*wmo 1423
This theorem is referenced by:  euxfr2 1973  2reuswap 1984
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
Copyright terms: Public domain