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Theorem rmo2 3423
Description: Alternate definition of restricted "at most one." Note that  E* x  e.  A ph is not equivalent to  E. y  e.  A A. x  e.  A ( ph  ->  x  =  y ) (in analogy to reu6 3288); to see this, let  A be the empty set. However, one direction of this pattern holds; see rmo2i 3424. (Contributed by NM, 17-Jun-2017.)
Hypothesis
Ref Expression
rmo2.1  |-  F/ y
ph
Assertion
Ref Expression
rmo2  |-  ( E* x  e.  A  ph  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem rmo2
StepHypRef Expression
1 df-rmo 2815 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
2 nfv 1708 . . . 4  |-  F/ y  x  e.  A
3 rmo2.1 . . . 4  |-  F/ y
ph
42, 3nfan 1929 . . 3  |-  F/ y ( x  e.  A  /\  ph )
54mo2 2294 . 2  |-  ( E* x ( x  e.  A  /\  ph )  <->  E. y A. x ( ( x  e.  A  /\  ph )  ->  x  =  y ) )
6 impexp 446 . . . . 5  |-  ( ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  ( x  e.  A  ->  ( ph  ->  x  =  y ) ) )
76albii 1641 . . . 4  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  A. x ( x  e.  A  ->  ( ph  ->  x  =  y ) ) )
8 df-ral 2812 . . . 4  |-  ( A. x  e.  A  ( ph  ->  x  =  y )  <->  A. x ( x  e.  A  ->  ( ph  ->  x  =  y ) ) )
97, 8bitr4i 252 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  A. x  e.  A  ( ph  ->  x  =  y ) )
109exbii 1668 . 2  |-  ( E. y A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
111, 5, 103bitri 271 1  |-  ( E* x  e.  A  ph  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393   E.wex 1613   F/wnf 1617    e. wcel 1819   E*wmo 2284   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:  rmo2i  3424  disjiun  4447  rmoeq  27513  rmoanim  32387
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