ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  equveli Unicode version

Theorem equveli 1689
Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1688.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
equveli  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )

Proof of Theorem equveli
StepHypRef Expression
1 albiim 1421 . 2  |-  ( A. z ( z  =  x  <->  z  =  y )  <->  ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) ) )
2 ax12or 1448 . . 3  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )
3 equequ1 1645 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  =  x  <->  x  =  x ) )
4 equequ1 1645 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
53, 4imbi12d 232 . . . . . . . 8  |-  ( z  =  x  ->  (
( z  =  x  ->  z  =  y )  <->  ( x  =  x  ->  x  =  y ) ) )
65sps 1475 . . . . . . 7  |-  ( A. z  z  =  x  ->  ( ( z  =  x  ->  z  =  y )  <->  ( x  =  x  ->  x  =  y ) ) )
76dral2 1666 . . . . . 6  |-  ( A. z  z  =  x  ->  ( A. z ( z  =  x  -> 
z  =  y )  <->  A. z ( x  =  x  ->  x  =  y ) ) )
8 equid 1634 . . . . . . . . 9  |-  x  =  x
98a1bi 241 . . . . . . . 8  |-  ( x  =  y  <->  ( x  =  x  ->  x  =  y ) )
109biimpri 131 . . . . . . 7  |-  ( ( x  =  x  ->  x  =  y )  ->  x  =  y )
1110sps 1475 . . . . . 6  |-  ( A. z ( x  =  x  ->  x  =  y )  ->  x  =  y )
127, 11syl6bi 161 . . . . 5  |-  ( A. z  z  =  x  ->  ( A. z ( z  =  x  -> 
z  =  y )  ->  x  =  y ) )
1312adantrd 273 . . . 4  |-  ( A. z  z  =  x  ->  ( ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) )  ->  x  =  y ) )
14 equequ1 1645 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  y  <->  y  =  y ) )
15 equequ1 1645 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  x  <->  y  =  x ) )
1614, 15imbi12d 232 . . . . . . . . 9  |-  ( z  =  y  ->  (
( z  =  y  ->  z  =  x )  <->  ( y  =  y  ->  y  =  x ) ) )
1716sps 1475 . . . . . . . 8  |-  ( A. z  z  =  y  ->  ( ( z  =  y  ->  z  =  x )  <->  ( y  =  y  ->  y  =  x ) ) )
1817dral1 1665 . . . . . . 7  |-  ( A. z  z  =  y  ->  ( A. z ( z  =  y  -> 
z  =  x )  <->  A. y ( y  =  y  ->  y  =  x ) ) )
19 equid 1634 . . . . . . . . 9  |-  y  =  y
20 ax-4 1445 . . . . . . . . 9  |-  ( A. y ( y  =  y  ->  y  =  x )  ->  (
y  =  y  -> 
y  =  x ) )
2119, 20mpi 15 . . . . . . . 8  |-  ( A. y ( y  =  y  ->  y  =  x )  ->  y  =  x )
22 equcomi 1637 . . . . . . . 8  |-  ( y  =  x  ->  x  =  y )
2321, 22syl 14 . . . . . . 7  |-  ( A. y ( y  =  y  ->  y  =  x )  ->  x  =  y )
2418, 23syl6bi 161 . . . . . 6  |-  ( A. z  z  =  y  ->  ( A. z ( z  =  y  -> 
z  =  x )  ->  x  =  y ) )
2524adantld 272 . . . . 5  |-  ( A. z  z  =  y  ->  ( ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) )  ->  x  =  y ) )
26 hba1 1478 . . . . . . . . . 10  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  A. z A. z
( x  =  y  ->  A. z  x  =  y ) )
27 hbequid 1451 . . . . . . . . . . 11  |-  ( x  =  x  ->  A. z  x  =  x )
2827a1i 9 . . . . . . . . . 10  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( x  =  x  ->  A. z  x  =  x ) )
29 ax-4 1445 . . . . . . . . . 10  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( x  =  y  ->  A. z  x  =  y ) )
3026, 28, 29hbimd 1510 . . . . . . . . 9  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( ( x  =  x  ->  x  =  y )  ->  A. z
( x  =  x  ->  x  =  y ) ) )
3130a5i 1480 . . . . . . . 8  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  A. z ( ( x  =  x  ->  x  =  y )  ->  A. z ( x  =  x  ->  x  =  y ) ) )
32 equtr 1642 . . . . . . . . . 10  |-  ( z  =  x  ->  (
x  =  x  -> 
z  =  x ) )
33 ax-8 1440 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
3432, 33imim12d 73 . . . . . . . . 9  |-  ( z  =  x  ->  (
( z  =  x  ->  z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) )
3534ax-gen 1383 . . . . . . . 8  |-  A. z
( z  =  x  ->  ( ( z  =  x  ->  z  =  y )  -> 
( x  =  x  ->  x  =  y ) ) )
36 19.26 1415 . . . . . . . . 9  |-  ( A. z ( ( ( x  =  x  ->  x  =  y )  ->  A. z ( x  =  x  ->  x  =  y ) )  /\  ( z  =  x  ->  ( (
z  =  x  -> 
z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) ) )  <->  ( A. z
( ( x  =  x  ->  x  =  y )  ->  A. z
( x  =  x  ->  x  =  y ) )  /\  A. z ( z  =  x  ->  ( (
z  =  x  -> 
z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) ) ) )
37 spimth 1670 . . . . . . . . 9  |-  ( A. z ( ( ( x  =  x  ->  x  =  y )  ->  A. z ( x  =  x  ->  x  =  y ) )  /\  ( z  =  x  ->  ( (
z  =  x  -> 
z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) ) )  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  (
x  =  x  ->  x  =  y )
) )
3836, 37sylbir 133 . . . . . . . 8  |-  ( ( A. z ( ( x  =  x  ->  x  =  y )  ->  A. z ( x  =  x  ->  x  =  y ) )  /\  A. z ( z  =  x  -> 
( ( z  =  x  ->  z  =  y )  ->  (
x  =  x  ->  x  =  y )
) ) )  -> 
( A. z ( z  =  x  -> 
z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) )
3931, 35, 38sylancl 404 . . . . . . 7  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( A. z ( z  =  x  -> 
z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) )
408, 39mpii 43 . . . . . 6  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( A. z ( z  =  x  -> 
z  =  y )  ->  x  =  y ) )
4140adantrd 273 . . . . 5  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) )  ->  x  =  y ) )
4225, 41jaoi 671 . . . 4  |-  ( ( A. z  z  =  y  \/  A. z
( x  =  y  ->  A. z  x  =  y ) )  -> 
( ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) )  ->  x  =  y ) )
4313, 42jaoi 671 . . 3  |-  ( ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )  ->  (
( A. z ( z  =  x  -> 
z  =  y )  /\  A. z ( z  =  y  -> 
z  =  x ) )  ->  x  =  y ) )
442, 43ax-mp 7 . 2  |-  ( ( A. z ( z  =  x  ->  z  =  y )  /\  A. z ( z  =  y  ->  z  =  x ) )  ->  x  =  y )
451, 44sylbi 119 1  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664   A.wal 1287    = wceq 1289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator