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Theorem equveli 1757
Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1756.) (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 1485 . 2  |-  ( A. z ( z  =  x  <->  z  =  y )  <->  ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) ) )
2 ax12or 1506 . . 3  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )
3 equequ1 1710 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  =  x  <->  x  =  x ) )
4 equequ1 1710 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
53, 4imbi12d 234 . . . . . . . 8  |-  ( z  =  x  ->  (
( z  =  x  ->  z  =  y )  <->  ( x  =  x  ->  x  =  y ) ) )
65sps 1535 . . . . . . 7  |-  ( A. z  z  =  x  ->  ( ( z  =  x  ->  z  =  y )  <->  ( x  =  x  ->  x  =  y ) ) )
76dral2 1729 . . . . . 6  |-  ( A. z  z  =  x  ->  ( A. z ( z  =  x  -> 
z  =  y )  <->  A. z ( x  =  x  ->  x  =  y ) ) )
8 equid 1699 . . . . . . . . 9  |-  x  =  x
98a1bi 243 . . . . . . . 8  |-  ( x  =  y  <->  ( x  =  x  ->  x  =  y ) )
109biimpri 133 . . . . . . 7  |-  ( ( x  =  x  ->  x  =  y )  ->  x  =  y )
1110sps 1535 . . . . . 6  |-  ( A. z ( x  =  x  ->  x  =  y )  ->  x  =  y )
127, 11syl6bi 163 . . . . 5  |-  ( A. z  z  =  x  ->  ( A. z ( z  =  x  -> 
z  =  y )  ->  x  =  y ) )
1312adantrd 279 . . . 4  |-  ( A. z  z  =  x  ->  ( ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) )  ->  x  =  y ) )
14 equequ1 1710 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  y  <->  y  =  y ) )
15 equequ1 1710 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  x  <->  y  =  x ) )
1614, 15imbi12d 234 . . . . . . . . 9  |-  ( z  =  y  ->  (
( z  =  y  ->  z  =  x )  <->  ( y  =  y  ->  y  =  x ) ) )
1716sps 1535 . . . . . . . 8  |-  ( A. z  z  =  y  ->  ( ( z  =  y  ->  z  =  x )  <->  ( y  =  y  ->  y  =  x ) ) )
1817dral1 1728 . . . . . . 7  |-  ( A. z  z  =  y  ->  ( A. z ( z  =  y  -> 
z  =  x )  <->  A. y ( y  =  y  ->  y  =  x ) ) )
19 equid 1699 . . . . . . . . 9  |-  y  =  y
20 ax-4 1508 . . . . . . . . 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 1702 . . . . . . . 8  |-  ( y  =  x  ->  x  =  y )
2321, 22syl 14 . . . . . . 7  |-  ( A. y ( y  =  y  ->  y  =  x )  ->  x  =  y )
2418, 23syl6bi 163 . . . . . 6  |-  ( A. z  z  =  y  ->  ( A. z ( z  =  y  -> 
z  =  x )  ->  x  =  y ) )
2524adantld 278 . . . . 5  |-  ( A. z  z  =  y  ->  ( ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) )  ->  x  =  y ) )
26 hba1 1538 . . . . . . . . . 10  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  A. z A. z
( x  =  y  ->  A. z  x  =  y ) )
27 hbequid 1511 . . . . . . . . . . 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 1508 . . . . . . . . . 10  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( x  =  y  ->  A. z  x  =  y ) )
3026, 28, 29hbimd 1571 . . . . . . . . 9  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( ( x  =  x  ->  x  =  y )  ->  A. z
( x  =  x  ->  x  =  y ) ) )
3130a5i 1541 . . . . . . . 8  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  A. z ( ( x  =  x  ->  x  =  y )  ->  A. z ( x  =  x  ->  x  =  y ) ) )
32 equtr 1707 . . . . . . . . . 10  |-  ( z  =  x  ->  (
x  =  x  -> 
z  =  x ) )
33 ax-8 1502 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
3432, 33imim12d 74 . . . . . . . . 9  |-  ( z  =  x  ->  (
( z  =  x  ->  z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) )
3534ax-gen 1447 . . . . . . . 8  |-  A. z
( z  =  x  ->  ( ( z  =  x  ->  z  =  y )  -> 
( x  =  x  ->  x  =  y ) ) )
36 19.26 1479 . . . . . . . . 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 1733 . . . . . . . . 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 135 . . . . . . . 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 413 . . . . . . 7  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( A. z ( z  =  x  -> 
z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) )
408, 39mpii 44 . . . . . 6  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( A. z ( z  =  x  -> 
z  =  y )  ->  x  =  y ) )
4140adantrd 279 . . . . 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 716 . . . 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 716 . . 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 5 . 2  |-  ( ( A. z ( z  =  x  ->  z  =  y )  /\  A. z ( z  =  y  ->  z  =  x ) )  ->  x  =  y )
451, 44sylbi 121 1  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708   A.wal 1351    = wceq 1353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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