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Theorem rmo4f 26922
Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
rmo4f.1  |-  F/_ x A
rmo4f.2  |-  F/_ y A
rmo4f.3  |-  F/ x ps
rmo4f.4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rmo4f  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  ps )  ->  x  =  y ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    A( x, y)

Proof of Theorem rmo4f
StepHypRef Expression
1 rmo4f.1 . . 3  |-  F/_ x A
2 rmo4f.2 . . 3  |-  F/_ y A
3 nfv 1678 . . 3  |-  F/ y
ph
41, 2, 3rmo3f 26920 . 2  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
5 rmo4f.3 . . . . . 6  |-  F/ x ps
6 rmo4f.4 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
75, 6sbie 2116 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
87anbi2i 694 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  <->  (
ph  /\  ps )
)
98imbi1i 325 . . 3  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  ps )  ->  x  =  y )
)
1092ralbii 2889 . 2  |-  ( A. x  e.  A  A. y  e.  A  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
114, 10bitri 249 1  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  ps )  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   F/wnf 1594   [wsb 1706   F/_wnfc 2608   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 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rmo 2815
This theorem is referenced by:  disjorf  26963  funcnv5mpt  27033
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