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Theorem euind 3123
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 1986 . . . . 5  |-  ( E. x ph  <->  E. y ps )
3 euind.1 . . . . . . . . 9  |-  B  e. 
_V
43isseti 2964 . . . . . . . 8  |-  E. z 
z  =  B
54biantrur 494 . . . . . . 7  |-  ( ps  <->  ( E. z  z  =  B  /\  ps )
)
65exbii 1593 . . . . . 6  |-  ( E. y ps  <->  E. y
( E. z  z  =  B  /\  ps ) )
7 19.41v 1925 . . . . . . 7  |-  ( E. z ( z  =  B  /\  ps )  <->  ( E. z  z  =  B  /\  ps )
)
87exbii 1593 . . . . . 6  |-  ( E. y E. z ( z  =  B  /\  ps )  <->  E. y ( E. z  z  =  B  /\  ps ) )
9 excom 1757 . . . . . 6  |-  ( E. y E. z ( z  =  B  /\  ps )  <->  E. z E. y
( z  =  B  /\  ps ) )
106, 8, 93bitr2i 266 . . . . 5  |-  ( E. y ps  <->  E. z E. y ( z  =  B  /\  ps )
)
112, 10bitri 242 . . . 4  |-  ( E. x ph  <->  E. z E. y ( z  =  B  /\  ps )
)
12 eqeq2 2447 . . . . . . . . 9  |-  ( A  =  B  ->  (
z  =  A  <->  z  =  B ) )
1312imim2i 14 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  ->  A  =  B )  ->  ( ( ph  /\  ps )  -> 
( z  =  A  <-> 
z  =  B ) ) )
14 bi2 191 . . . . . . . . . 10  |-  ( ( z  =  A  <->  z  =  B )  ->  (
z  =  B  -> 
z  =  A ) )
1514imim2i 14 . . . . . . . . 9  |-  ( ( ( ph  /\  ps )  ->  ( z  =  A  <->  z  =  B ) )  ->  (
( ph  /\  ps )  ->  ( z  =  B  ->  z  =  A ) ) )
16 an31 777 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ps )  /\  z  =  B )  <->  ( ( z  =  B  /\  ps )  /\  ph ) )
1716imbi1i 317 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ps )  /\  z  =  B )  ->  z  =  A )  <->  ( (
( z  =  B  /\  ps )  /\  ph )  ->  z  =  A ) )
18 impexp 435 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ps )  /\  z  =  B )  ->  z  =  A )  <->  ( ( ph  /\  ps )  -> 
( z  =  B  ->  z  =  A ) ) )
19 impexp 435 . . . . . . . . . 10  |-  ( ( ( ( z  =  B  /\  ps )  /\  ph )  ->  z  =  A )  <->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2017, 18, 193bitr3i 268 . . . . . . . . 9  |-  ( ( ( ph  /\  ps )  ->  ( z  =  B  ->  z  =  A ) )  <->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2115, 20sylib 190 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  ->  ( z  =  A  <->  z  =  B ) )  ->  (
( z  =  B  /\  ps )  -> 
( ph  ->  z  =  A ) ) )
2213, 21syl 16 . . . . . . 7  |-  ( ( ( ph  /\  ps )  ->  A  =  B )  ->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
23222alimi 1570 . . . . . 6  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  A. x A. y
( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
24 19.23v 1915 . . . . . . . 8  |-  ( A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <-> 
( E. y ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2524albii 1576 . . . . . . 7  |-  ( A. x A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <->  A. x ( E. y ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
26 19.21v 1914 . . . . . . 7  |-  ( A. x ( E. y
( z  =  B  /\  ps )  -> 
( ph  ->  z  =  A ) )  <->  ( E. y ( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2725, 26bitri 242 . . . . . 6  |-  ( A. x A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <->  ( E. y
( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2823, 27sylib 190 . . . . 5  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. y
( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2928eximdv 1633 . . . 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 210 . . 3  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. x ph  ->  E. z A. x
( ph  ->  z  =  A ) ) )
3130imp 420 . 2  |-  ( ( A. x A. y
( ( ph  /\  ps )  ->  A  =  B )  /\  E. x ph )  ->  E. z A. x ( ph  ->  z  =  A ) )
32 pm4.24 626 . . . . . . . 8  |-  ( ph  <->  (
ph  /\  ph ) )
3332biimpi 188 . . . . . . 7  |-  ( ph  ->  ( ph  /\  ph ) )
34 prth 556 . . . . . . 7  |-  ( ( ( ph  ->  z  =  A )  /\  ( ph  ->  w  =  A ) )  ->  (
( ph  /\  ph )  ->  ( z  =  A  /\  w  =  A ) ) )
35 eqtr3 2457 . . . . . . 7  |-  ( ( z  =  A  /\  w  =  A )  ->  z  =  w )
3633, 34, 35syl56 33 . . . . . 6  |-  ( ( ( ph  ->  z  =  A )  /\  ( ph  ->  w  =  A ) )  ->  ( ph  ->  z  =  w ) )
3736alanimi 1572 . . . . 5  |-  ( ( A. x ( ph  ->  z  =  A )  /\  A. x (
ph  ->  w  =  A ) )  ->  A. x
( ph  ->  z  =  w ) )
38 19.23v 1915 . . . . . . 7  |-  ( A. x ( ph  ->  z  =  w )  <->  ( E. x ph  ->  z  =  w ) )
3938biimpi 188 . . . . . 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 31 . . . 4  |-  ( E. x ph  ->  (
( A. x (
ph  ->  z  =  A )  /\  A. x
( ph  ->  w  =  A ) )  -> 
z  =  w ) )
4241alrimivv 1643 . . 3  |-  ( E. x ph  ->  A. z A. w ( ( A. x ( ph  ->  z  =  A )  /\  A. x ( ph  ->  w  =  A ) )  ->  z  =  w ) )
4342adantl 454 . 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 2444 . . . . 5  |-  ( z  =  w  ->  (
z  =  A  <->  w  =  A ) )
4544imbi2d 309 . . . 4  |-  ( z  =  w  ->  (
( ph  ->  z  =  A )  <->  ( ph  ->  w  =  A ) ) )
4645albidv 1636 . . 3  |-  ( z  =  w  ->  ( A. x ( ph  ->  z  =  A )  <->  A. x
( ph  ->  w  =  A ) ) )
4746eu4 2322 . 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 647 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    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   E!weu 2283   _Vcvv 2958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
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