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Theorem euind 2751
Description: Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
Hypotheses
Ref Expression
euind.1  |-  B  e. 
_V
euind.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
euind.3  |-  ( x  =  y  ->  A  =  B )
Assertion
Ref Expression
euind  |-  ( ( A. x A. y
( ( ph  /\  ps )  ->  A  =  B )  /\  E. x ph )  ->  E! z A. x ( ph  ->  z  =  A ) )
Distinct variable groups:    y, z, ph    x, z, ps    y, A, z    x, B, z    x, y
Allowed substitution hints:    ph( x)    ps( y)    A( x)    B( y)

Proof of Theorem euind
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 euind.2 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21cbvexv 1811 . . . . 5  |-  ( E. x ph  <->  E. y ps )
3 euind.1 . . . . . . . . 9  |-  B  e. 
_V
43isseti 2580 . . . . . . . 8  |-  E. z 
z  =  B
54biantrur 291 . . . . . . 7  |-  ( ps  <->  ( E. z  z  =  B  /\  ps )
)
65exbii 1512 . . . . . 6  |-  ( E. y ps  <->  E. y
( E. z  z  =  B  /\  ps ) )
7 19.41v 1798 . . . . . . 7  |-  ( E. z ( z  =  B  /\  ps )  <->  ( E. z  z  =  B  /\  ps )
)
87exbii 1512 . . . . . 6  |-  ( E. y E. z ( z  =  B  /\  ps )  <->  E. y ( E. z  z  =  B  /\  ps ) )
9 excom 1570 . . . . . 6  |-  ( E. y E. z ( z  =  B  /\  ps )  <->  E. z E. y
( z  =  B  /\  ps ) )
106, 8, 93bitr2i 201 . . . . 5  |-  ( E. y ps  <->  E. z E. y ( z  =  B  /\  ps )
)
112, 10bitri 177 . . . 4  |-  ( E. x ph  <->  E. z E. y ( z  =  B  /\  ps )
)
12 eqeq2 2065 . . . . . . . . 9  |-  ( A  =  B  ->  (
z  =  A  <->  z  =  B ) )
1312imim2i 12 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  ->  A  =  B )  ->  ( ( ph  /\  ps )  -> 
( z  =  A  <-> 
z  =  B ) ) )
14 bi2 125 . . . . . . . . . 10  |-  ( ( z  =  A  <->  z  =  B )  ->  (
z  =  B  -> 
z  =  A ) )
1514imim2i 12 . . . . . . . . 9  |-  ( ( ( ph  /\  ps )  ->  ( z  =  A  <->  z  =  B ) )  ->  (
( ph  /\  ps )  ->  ( z  =  B  ->  z  =  A ) ) )
16 an31 506 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ps )  /\  z  =  B )  <->  ( ( z  =  B  /\  ps )  /\  ph ) )
1716imbi1i 231 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ps )  /\  z  =  B )  ->  z  =  A )  <->  ( (
( z  =  B  /\  ps )  /\  ph )  ->  z  =  A ) )
18 impexp 254 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ps )  /\  z  =  B )  ->  z  =  A )  <->  ( ( ph  /\  ps )  -> 
( z  =  B  ->  z  =  A ) ) )
19 impexp 254 . . . . . . . . . 10  |-  ( ( ( ( z  =  B  /\  ps )  /\  ph )  ->  z  =  A )  <->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2017, 18, 193bitr3i 203 . . . . . . . . 9  |-  ( ( ( ph  /\  ps )  ->  ( z  =  B  ->  z  =  A ) )  <->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2115, 20sylib 131 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  ->  ( z  =  A  <->  z  =  B ) )  ->  (
( z  =  B  /\  ps )  -> 
( ph  ->  z  =  A ) ) )
2213, 21syl 14 . . . . . . 7  |-  ( ( ( ph  /\  ps )  ->  A  =  B )  ->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
23222alimi 1361 . . . . . 6  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  A. x A. y
( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
24 19.23v 1779 . . . . . . . 8  |-  ( A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <-> 
( E. y ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2524albii 1375 . . . . . . 7  |-  ( A. x A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <->  A. x ( E. y ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
26 19.21v 1769 . . . . . . 7  |-  ( A. x ( E. y
( z  =  B  /\  ps )  -> 
( ph  ->  z  =  A ) )  <->  ( E. y ( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2725, 26bitri 177 . . . . . 6  |-  ( A. x A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <->  ( E. y
( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2823, 27sylib 131 . . . . 5  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. y
( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2928eximdv 1776 . . . 4  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. z E. y ( z  =  B  /\  ps )  ->  E. z A. x
( ph  ->  z  =  A ) ) )
3011, 29syl5bi 145 . . 3  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. x ph  ->  E. z A. x
( ph  ->  z  =  A ) ) )
3130imp 119 . 2  |-  ( ( A. x A. y
( ( ph  /\  ps )  ->  A  =  B )  /\  E. x ph )  ->  E. z A. x ( ph  ->  z  =  A ) )
32 pm4.24 381 . . . . . . . 8  |-  ( ph  <->  (
ph  /\  ph ) )
3332biimpi 117 . . . . . . 7  |-  ( ph  ->  ( ph  /\  ph ) )
34 prth 330 . . . . . . 7  |-  ( ( ( ph  ->  z  =  A )  /\  ( ph  ->  w  =  A ) )  ->  (
( ph  /\  ph )  ->  ( z  =  A  /\  w  =  A ) ) )
35 eqtr3 2075 . . . . . . 7  |-  ( ( z  =  A  /\  w  =  A )  ->  z  =  w )
3633, 34, 35syl56 34 . . . . . 6  |-  ( ( ( ph  ->  z  =  A )  /\  ( ph  ->  w  =  A ) )  ->  ( ph  ->  z  =  w ) )
3736alanimi 1364 . . . . 5  |-  ( ( A. x ( ph  ->  z  =  A )  /\  A. x (
ph  ->  w  =  A ) )  ->  A. x
( ph  ->  z  =  w ) )
38 19.23v 1779 . . . . . . 7  |-  ( A. x ( ph  ->  z  =  w )  <->  ( E. x ph  ->  z  =  w ) )
3938biimpi 117 . . . . . 6  |-  ( A. x ( ph  ->  z  =  w )  -> 
( E. x ph  ->  z  =  w ) )
4039com12 30 . . . . 5  |-  ( E. x ph  ->  ( A. x ( ph  ->  z  =  w )  -> 
z  =  w ) )
4137, 40syl5 32 . . . 4  |-  ( E. x ph  ->  (
( A. x (
ph  ->  z  =  A )  /\  A. x
( ph  ->  w  =  A ) )  -> 
z  =  w ) )
4241alrimivv 1771 . . 3  |-  ( E. x ph  ->  A. z A. w ( ( A. x ( ph  ->  z  =  A )  /\  A. x ( ph  ->  w  =  A ) )  ->  z  =  w ) )
4342adantl 266 . 2  |-  ( ( A. x A. y
( ( ph  /\  ps )  ->  A  =  B )  /\  E. x ph )  ->  A. z A. w ( ( A. x ( ph  ->  z  =  A )  /\  A. x ( ph  ->  w  =  A ) )  ->  z  =  w ) )
44 eqeq1 2062 . . . . 5  |-  ( z  =  w  ->  (
z  =  A  <->  w  =  A ) )
4544imbi2d 223 . . . 4  |-  ( z  =  w  ->  (
( ph  ->  z  =  A )  <->  ( ph  ->  w  =  A ) ) )
4645albidv 1721 . . 3  |-  ( z  =  w  ->  ( A. x ( ph  ->  z  =  A )  <->  A. x
( ph  ->  w  =  A ) ) )
4746eu4 1978 . 2  |-  ( E! z A. x (
ph  ->  z  =  A )  <->  ( E. z A. x ( ph  ->  z  =  A )  /\  A. z A. w ( ( A. x (
ph  ->  z  =  A )  /\  A. x
( ph  ->  w  =  A ) )  -> 
z  =  w ) ) )
4831, 43, 47sylanbrc 402 1  |-  ( ( A. x A. y
( ( ph  /\  ps )  ->  A  =  B )  /\  E. x ph )  ->  E! z A. x ( ph  ->  z  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   E!weu 1916   _Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by: (None)
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