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Theorem euind 3225
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 2117 . . . . 5  |-  ( E. x ph  <->  E. y ps )
3 euind.1 . . . . . . . . 9  |-  B  e. 
_V
43isseti 3051 . . . . . . . 8  |-  E. z 
z  =  B
54biantrur 509 . . . . . . 7  |-  ( ps  <->  ( E. z  z  =  B  /\  ps )
)
65exbii 1718 . . . . . 6  |-  ( E. y ps  <->  E. y
( E. z  z  =  B  /\  ps ) )
7 19.41v 1830 . . . . . . 7  |-  ( E. z ( z  =  B  /\  ps )  <->  ( E. z  z  =  B  /\  ps )
)
87exbii 1718 . . . . . 6  |-  ( E. y E. z ( z  =  B  /\  ps )  <->  E. y ( E. z  z  =  B  /\  ps ) )
9 excom 1927 . . . . . 6  |-  ( E. y E. z ( z  =  B  /\  ps )  <->  E. z E. y
( z  =  B  /\  ps ) )
106, 8, 93bitr2i 277 . . . . 5  |-  ( E. y ps  <->  E. z E. y ( z  =  B  /\  ps )
)
112, 10bitri 253 . . . 4  |-  ( E. x ph  <->  E. z E. y ( z  =  B  /\  ps )
)
12 eqeq2 2462 . . . . . . . . 9  |-  ( A  =  B  ->  (
z  =  A  <->  z  =  B ) )
1312imim2i 16 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  ->  A  =  B )  ->  ( ( ph  /\  ps )  -> 
( z  =  A  <-> 
z  =  B ) ) )
14 biimpr 202 . . . . . . . . . 10  |-  ( ( z  =  A  <->  z  =  B )  ->  (
z  =  B  -> 
z  =  A ) )
1514imim2i 16 . . . . . . . . 9  |-  ( ( ( ph  /\  ps )  ->  ( z  =  A  <->  z  =  B ) )  ->  (
( ph  /\  ps )  ->  ( z  =  B  ->  z  =  A ) ) )
16 an31 809 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ps )  /\  z  =  B )  <->  ( ( z  =  B  /\  ps )  /\  ph ) )
1716imbi1i 327 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ps )  /\  z  =  B )  ->  z  =  A )  <->  ( (
( z  =  B  /\  ps )  /\  ph )  ->  z  =  A ) )
18 impexp 448 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ps )  /\  z  =  B )  ->  z  =  A )  <->  ( ( ph  /\  ps )  -> 
( z  =  B  ->  z  =  A ) ) )
19 impexp 448 . . . . . . . . . 10  |-  ( ( ( ( z  =  B  /\  ps )  /\  ph )  ->  z  =  A )  <->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2017, 18, 193bitr3i 279 . . . . . . . . 9  |-  ( ( ( ph  /\  ps )  ->  ( z  =  B  ->  z  =  A ) )  <->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2115, 20sylib 200 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  ->  ( z  =  A  <->  z  =  B ) )  ->  (
( z  =  B  /\  ps )  -> 
( ph  ->  z  =  A ) ) )
2213, 21syl 17 . . . . . . 7  |-  ( ( ( ph  /\  ps )  ->  A  =  B )  ->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
23222alimi 1685 . . . . . 6  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  A. x A. y
( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
24 19.23v 1818 . . . . . . . 8  |-  ( A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <-> 
( E. y ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2524albii 1691 . . . . . . 7  |-  ( A. x A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <->  A. x ( E. y ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
26 19.21v 1786 . . . . . . 7  |-  ( A. x ( E. y
( z  =  B  /\  ps )  -> 
( ph  ->  z  =  A ) )  <->  ( E. y ( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2725, 26bitri 253 . . . . . 6  |-  ( A. x A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <->  ( E. y
( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2823, 27sylib 200 . . . . 5  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. y
( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2928eximdv 1764 . . . 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 221 . . 3  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. x ph  ->  E. z A. x
( ph  ->  z  =  A ) ) )
3130imp 431 . 2  |-  ( ( A. x A. y
( ( ph  /\  ps )  ->  A  =  B )  /\  E. x ph )  ->  E. z A. x ( ph  ->  z  =  A ) )
32 pm4.24 649 . . . . . . . 8  |-  ( ph  <->  (
ph  /\  ph ) )
3332biimpi 198 . . . . . . 7  |-  ( ph  ->  ( ph  /\  ph ) )
34 prth 575 . . . . . . 7  |-  ( ( ( ph  ->  z  =  A )  /\  ( ph  ->  w  =  A ) )  ->  (
( ph  /\  ph )  ->  ( z  =  A  /\  w  =  A ) ) )
35 eqtr3 2472 . . . . . . 7  |-  ( ( z  =  A  /\  w  =  A )  ->  z  =  w )
3633, 34, 35syl56 35 . . . . . 6  |-  ( ( ( ph  ->  z  =  A )  /\  ( ph  ->  w  =  A ) )  ->  ( ph  ->  z  =  w ) )
3736alanimi 1688 . . . . 5  |-  ( ( A. x ( ph  ->  z  =  A )  /\  A. x (
ph  ->  w  =  A ) )  ->  A. x
( ph  ->  z  =  w ) )
38 19.23v 1818 . . . . . . 7  |-  ( A. x ( ph  ->  z  =  w )  <->  ( E. x ph  ->  z  =  w ) )
3938biimpi 198 . . . . . 6  |-  ( A. x ( ph  ->  z  =  w )  -> 
( E. x ph  ->  z  =  w ) )
4039com12 32 . . . . 5  |-  ( E. x ph  ->  ( A. x ( ph  ->  z  =  w )  -> 
z  =  w ) )
4137, 40syl5 33 . . . 4  |-  ( E. x ph  ->  (
( A. x (
ph  ->  z  =  A )  /\  A. x
( ph  ->  w  =  A ) )  -> 
z  =  w ) )
4241alrimivv 1774 . . 3  |-  ( E. x ph  ->  A. z A. w ( ( A. x ( ph  ->  z  =  A )  /\  A. x ( ph  ->  w  =  A ) )  ->  z  =  w ) )
4342adantl 468 . 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 2455 . . . . 5  |-  ( z  =  w  ->  (
z  =  A  <->  w  =  A ) )
4544imbi2d 318 . . . 4  |-  ( z  =  w  ->  (
( ph  ->  z  =  A )  <->  ( ph  ->  w  =  A ) ) )
4645albidv 1767 . . 3  |-  ( z  =  w  ->  ( A. x ( ph  ->  z  =  A )  <->  A. x
( ph  ->  w  =  A ) ) )
4746eu4 2347 . 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 670 1  |-  ( ( A. x A. y
( ( ph  /\  ps )  ->  A  =  B )  /\  E. x ph )  ->  E! z A. x ( ph  ->  z  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887   E!weu 2299   _Vcvv 3045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3047
This theorem is referenced by: (None)
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