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Theorem euind 2952
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 1943 . . . . 5  |-  ( E. x ph  <->  E. y ps )
3 euind.1 . . . . . . . . 9  |-  B  e. 
_V
43isseti 2794 . . . . . . . 8  |-  E. z 
z  =  B
54biantrur 492 . . . . . . 7  |-  ( ps  <->  ( E. z  z  =  B  /\  ps )
)
65exbii 1569 . . . . . 6  |-  ( E. y ps  <->  E. y
( E. z  z  =  B  /\  ps ) )
7 19.41v 1842 . . . . . . . 8  |-  ( E. z ( z  =  B  /\  ps )  <->  ( E. z  z  =  B  /\  ps )
)
87exbii 1569 . . . . . . 7  |-  ( E. y E. z ( z  =  B  /\  ps )  <->  E. y ( E. z  z  =  B  /\  ps ) )
9 excom 1786 . . . . . . 7  |-  ( E. y E. z ( z  =  B  /\  ps )  <->  E. z E. y
( z  =  B  /\  ps ) )
108, 9bitr3i 242 . . . . . 6  |-  ( E. y ( E. z 
z  =  B  /\  ps )  <->  E. z E. y
( z  =  B  /\  ps ) )
116, 10bitri 240 . . . . 5  |-  ( E. y ps  <->  E. z E. y ( z  =  B  /\  ps )
)
122, 11bitri 240 . . . 4  |-  ( E. x ph  <->  E. z E. y ( z  =  B  /\  ps )
)
13 eqeq2 2292 . . . . . . . . 9  |-  ( A  =  B  ->  (
z  =  A  <->  z  =  B ) )
1413imim2i 13 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  ->  A  =  B )  ->  ( ( ph  /\  ps )  -> 
( z  =  A  <-> 
z  =  B ) ) )
15 bi2 189 . . . . . . . . . 10  |-  ( ( z  =  A  <->  z  =  B )  ->  (
z  =  B  -> 
z  =  A ) )
1615imim2i 13 . . . . . . . . 9  |-  ( ( ( ph  /\  ps )  ->  ( z  =  A  <->  z  =  B ) )  ->  (
( ph  /\  ps )  ->  ( z  =  B  ->  z  =  A ) ) )
17 an31 775 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ps )  /\  z  =  B )  <->  ( ( z  =  B  /\  ps )  /\  ph ) )
1817imbi1i 315 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ps )  /\  z  =  B )  ->  z  =  A )  <->  ( (
( z  =  B  /\  ps )  /\  ph )  ->  z  =  A ) )
19 impexp 433 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ps )  /\  z  =  B )  ->  z  =  A )  <->  ( ( ph  /\  ps )  -> 
( z  =  B  ->  z  =  A ) ) )
20 impexp 433 . . . . . . . . . 10  |-  ( ( ( ( z  =  B  /\  ps )  /\  ph )  ->  z  =  A )  <->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2118, 19, 203bitr3i 266 . . . . . . . . 9  |-  ( ( ( ph  /\  ps )  ->  ( z  =  B  ->  z  =  A ) )  <->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2216, 21sylib 188 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  ->  ( z  =  A  <->  z  =  B ) )  ->  (
( z  =  B  /\  ps )  -> 
( ph  ->  z  =  A ) ) )
2314, 22syl 15 . . . . . . 7  |-  ( ( ( ph  /\  ps )  ->  A  =  B )  ->  ( (
z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
24232alimi 1547 . . . . . 6  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  A. x A. y
( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
25 19.23v 1832 . . . . . . . 8  |-  ( A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <-> 
( E. y ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
2625albii 1553 . . . . . . 7  |-  ( A. x A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <->  A. x ( E. y ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) ) )
27 19.21v 1831 . . . . . . 7  |-  ( A. x ( E. y
( z  =  B  /\  ps )  -> 
( ph  ->  z  =  A ) )  <->  ( E. y ( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2826, 27bitri 240 . . . . . 6  |-  ( A. x A. y ( ( z  =  B  /\  ps )  ->  ( ph  ->  z  =  A ) )  <->  ( E. y
( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
2924, 28sylib 188 . . . . 5  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. y
( z  =  B  /\  ps )  ->  A. x ( ph  ->  z  =  A ) ) )
3029eximdv 1608 . . . 4  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. z E. y ( z  =  B  /\  ps )  ->  E. z A. x
( ph  ->  z  =  A ) ) )
3112, 30syl5bi 208 . . 3  |-  ( A. x A. y ( (
ph  /\  ps )  ->  A  =  B )  ->  ( E. x ph  ->  E. z A. x
( ph  ->  z  =  A ) ) )
3231imp 418 . 2  |-  ( ( A. x A. y
( ( ph  /\  ps )  ->  A  =  B )  /\  E. x ph )  ->  E. z A. x ( ph  ->  z  =  A ) )
33 pm4.24 624 . . . . . . . 8  |-  ( ph  <->  (
ph  /\  ph ) )
3433biimpi 186 . . . . . . 7  |-  ( ph  ->  ( ph  /\  ph ) )
35 prth 554 . . . . . . 7  |-  ( ( ( ph  ->  z  =  A )  /\  ( ph  ->  w  =  A ) )  ->  (
( ph  /\  ph )  ->  ( z  =  A  /\  w  =  A ) ) )
36 eqtr3 2302 . . . . . . 7  |-  ( ( z  =  A  /\  w  =  A )  ->  z  =  w )
3734, 35, 36syl56 30 . . . . . 6  |-  ( ( ( ph  ->  z  =  A )  /\  ( ph  ->  w  =  A ) )  ->  ( ph  ->  z  =  w ) )
3837alanimi 1549 . . . . 5  |-  ( ( A. x ( ph  ->  z  =  A )  /\  A. x (
ph  ->  w  =  A ) )  ->  A. x
( ph  ->  z  =  w ) )
39 19.23v 1832 . . . . . . 7  |-  ( A. x ( ph  ->  z  =  w )  <->  ( E. x ph  ->  z  =  w ) )
4039biimpi 186 . . . . . 6  |-  ( A. x ( ph  ->  z  =  w )  -> 
( E. x ph  ->  z  =  w ) )
4140com12 27 . . . . 5  |-  ( E. x ph  ->  ( A. x ( ph  ->  z  =  w )  -> 
z  =  w ) )
4238, 41syl5 28 . . . 4  |-  ( E. x ph  ->  (
( A. x (
ph  ->  z  =  A )  /\  A. x
( ph  ->  w  =  A ) )  -> 
z  =  w ) )
4342alrimivv 1618 . . 3  |-  ( E. x ph  ->  A. z A. w ( ( A. x ( ph  ->  z  =  A )  /\  A. x ( ph  ->  w  =  A ) )  ->  z  =  w ) )
4443adantl 452 . 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 ) )
45 eqeq1 2289 . . . . 5  |-  ( z  =  w  ->  (
z  =  A  <->  w  =  A ) )
4645imbi2d 307 . . . 4  |-  ( z  =  w  ->  (
( ph  ->  z  =  A )  <->  ( ph  ->  w  =  A ) ) )
4746albidv 1611 . . 3  |-  ( z  =  w  ->  ( A. x ( ph  ->  z  =  A )  <->  A. x
( ph  ->  w  =  A ) ) )
4847eu4 2182 . 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 ) ) )
4932, 44, 48sylanbrc 645 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 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   _Vcvv 2788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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