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Theorem sb8mo 2175
Description: Variable substitution in uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
sb8eu.1  |-  F/ y
ph
Assertion
Ref Expression
sb8mo  |-  ( E* x ph  <->  E* y [ y  /  x ] ph )

Proof of Theorem sb8mo
StepHypRef Expression
1 sb8eu.1 . . . 4  |-  F/ y
ph
21sb8e 2046 . . 3  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
31sb8eu 2174 . . 3  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
42, 3imbi12i 316 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
5 df-mo 2161 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
6 df-mo 2161 . 2  |-  ( E* y [ y  /  x ] ph  <->  ( E. y [ y  /  x ] ph  ->  E! y [ y  /  x ] ph ) )
74, 5, 63bitr4i 268 1  |-  ( E* x ph  <->  E* y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1531   F/wnf 1534   [wsb 1638   E!weu 2156   E*wmo 2157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161
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