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Theorem rmoanim 32387
Description: Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2350. (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 446 . . . . 5  |-  ( ( ( ph  /\  ps )  ->  x  =  y )  <->  ( ph  ->  ( ps  ->  x  =  y ) ) )
21ralbii 2888 . . . 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 2856 . . . 4  |-  ( A. x  e.  A  ( ph  ->  ( ps  ->  x  =  y ) )  <-> 
( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) ) )
52, 4bitri 249 . . 3  |-  ( A. x  e.  A  (
( ph  /\  ps )  ->  x  =  y )  <-> 
( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) ) )
65exbii 1668 . 2  |-  ( E. y A. x  e.  A  ( ( ph  /\ 
ps )  ->  x  =  y )  <->  E. y
( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) ) )
7 nfv 1708 . . 3  |-  F/ y ( ph  /\  ps )
87rmo2 3423 . 2  |-  ( E* x  e.  A  (
ph  /\  ps )  <->  E. y A. x  e.  A  ( ( ph  /\ 
ps )  ->  x  =  y ) )
9 nfv 1708 . . . . 5  |-  F/ y ps
109rmo2 3423 . . . 4  |-  ( E* x  e.  A  ps  <->  E. y A. x  e.  A  ( ps  ->  x  =  y ) )
1110imbi2i 312 . . 3  |-  ( (
ph  ->  E* x  e.  A  ps )  <->  ( ph  ->  E. y A. x  e.  A  ( ps  ->  x  =  y ) ) )
12 19.37v 1769 . . 3  |-  ( E. y ( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) )  <->  ( ph  ->  E. y A. x  e.  A  ( ps  ->  x  =  y ) ) )
1311, 12bitr4i 252 . 2  |-  ( (
ph  ->  E* x  e.  A  ps )  <->  E. y
( ph  ->  A. x  e.  A  ( ps  ->  x  =  y ) ) )
146, 8, 133bitr4i 277 1  |-  ( E* x  e.  A  (
ph  /\  ps )  <->  (
ph  ->  E* x  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1613   F/wnf 1617   A.wral 2807   E*wrmo 2810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-nf 1618  df-eu 2287  df-mo 2288  df-ral 2812  df-rmo 2815
This theorem is referenced by:  2reu1  32394
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