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Theorem rmoanim 27924
Description: Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2336. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Hypothesis
Ref Expression
rmoanim.1  |-  F/ x ph
Assertion
Ref Expression
rmoanim  |-  ( E* x  e.  A (
ph  /\  ps )  <->  (
ph  ->  E* x  e.  A ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rmoanim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 impexp 434 . . . . 5  |-  ( ( ( ph  /\  ps )  ->  x  =  y )  <->  ( ph  ->  ( ps  ->  x  =  y ) ) )
21ralbii 2721 . . . 4  |-  ( A. x  e.  A  (
( ph  /\  ps )  ->  x  =  y )  <->  A. x  e.  A  ( ph  ->  ( ps  ->  x  =  y ) ) )
3 rmoanim.1 . . . . 5  |-  F/ x ph
43r19.21 2784 . . . 4  |-  ( A. x  e.  A  ( ph  ->  ( ps  ->  x  =  y ) )  <-> 
( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) ) )
52, 4bitri 241 . . 3  |-  ( A. x  e.  A  (
( ph  /\  ps )  ->  x  =  y )  <-> 
( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) ) )
65exbii 1592 . 2  |-  ( E. y A. x  e.  A  ( ( ph  /\ 
ps )  ->  x  =  y )  <->  E. y
( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) ) )
7 nfv 1629 . . 3  |-  F/ y ( ph  /\  ps )
87rmo2 3238 . 2  |-  ( E* x  e.  A (
ph  /\  ps )  <->  E. y A. x  e.  A  ( ( ph  /\ 
ps )  ->  x  =  y ) )
9 nfv 1629 . . . . 5  |-  F/ y ps
109rmo2 3238 . . . 4  |-  ( E* x  e.  A ps  <->  E. y A. x  e.  A  ( ps  ->  x  =  y ) )
1110imbi2i 304 . . 3  |-  ( (
ph  ->  E* x  e.  A ps )  <->  ( ph  ->  E. y A. x  e.  A  ( ps  ->  x  =  y ) ) )
12 19.37v 1922 . . 3  |-  ( E. y ( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) )  <->  ( ph  ->  E. y A. x  e.  A  ( ps  ->  x  =  y ) ) )
1311, 12bitr4i 244 . 2  |-  ( (
ph  ->  E* x  e.  A ps )  <->  E. y
( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) ) )
146, 8, 133bitr4i 269 1  |-  ( E* x  e.  A (
ph  /\  ps )  <->  (
ph  ->  E* x  e.  A ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550   F/wnf 1553   A.wral 2697   E*wrmo 2700
This theorem is referenced by:  2reu1  27931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-ral 2702  df-rmo 2705
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