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Theorem a12lem1 29752
Description: Proof of first hypothesis of a12study 29754. (Contributed by NM, 15-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a12lem1  |-  ( -. 
A. z  z  =  y  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y ) )

Proof of Theorem a12lem1
StepHypRef Expression
1 equequ1 1667 . . . . . . 7  |-  ( z  =  x  ->  (
z  =  x  <->  x  =  x ) )
2 equequ1 1667 . . . . . . 7  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
31, 2imbi12d 311 . . . . . 6  |-  ( z  =  x  ->  (
( z  =  x  ->  z  =  y )  <->  ( x  =  x  ->  x  =  y ) ) )
43sps 1751 . . . . 5  |-  ( A. z  z  =  x  ->  ( ( z  =  x  ->  z  =  y )  <->  ( x  =  x  ->  x  =  y ) ) )
54dral2 1919 . . . 4  |-  ( A. z  z  =  x  ->  ( A. z ( z  =  x  -> 
z  =  y )  <->  A. z ( x  =  x  ->  x  =  y ) ) )
6 equid 1662 . . . . . . 7  |-  x  =  x
76a1bi 327 . . . . . 6  |-  ( x  =  y  <->  ( x  =  x  ->  x  =  y ) )
87biimpri 197 . . . . 5  |-  ( ( x  =  x  ->  x  =  y )  ->  x  =  y )
98sps 1751 . . . 4  |-  ( A. z ( x  =  x  ->  x  =  y )  ->  x  =  y )
105, 9syl6bi 219 . . 3  |-  ( A. z  z  =  x  ->  ( A. z ( z  =  x  -> 
z  =  y )  ->  x  =  y ) )
1110a1d 22 . 2  |-  ( A. z  z  =  x  ->  ( -.  A. z 
z  =  y  -> 
( A. z ( z  =  x  -> 
z  =  y )  ->  x  =  y ) ) )
126nfth 1543 . . . . . . 7  |-  F/ z  x  =  x
1312a1i 10 . . . . . 6  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  x )
14 nfnae 1909 . . . . . . . . 9  |-  F/ z  -.  A. z  z  =  x
15 nfnae 1909 . . . . . . . . 9  |-  F/ z  -.  A. z  z  =  y
1614, 15nfan 1783 . . . . . . . 8  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
1716nfri 1754 . . . . . . 7  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  A. z
( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y ) )
18 ax12o 1887 . . . . . . . 8  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
1918imp 418 . . . . . . 7  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z  x  =  y )
)
2017, 19nfdh 1759 . . . . . 6  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
2113, 20nfimd 1773 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z
( x  =  x  ->  x  =  y ) )
22 equtr 1671 . . . . . . 7  |-  ( z  =  x  ->  (
x  =  x  -> 
z  =  x ) )
23 ax-8 1661 . . . . . . 7  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
2422, 23imim12d 68 . . . . . 6  |-  ( z  =  x  ->  (
( z  =  x  ->  z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) )
2524ax-gen 1536 . . . . 5  |-  A. z
( z  =  x  ->  ( ( z  =  x  ->  z  =  y )  -> 
( x  =  x  ->  x  =  y ) ) )
26 spimt 1927 . . . . 5  |-  ( ( F/ 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 )
) )
2721, 25, 26sylancl 643 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  (
x  =  x  ->  x  =  y )
) )
286, 27mpii 39 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y ) )
2928ex 423 . 2  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( A. z
( z  =  x  ->  z  =  y )  ->  x  =  y ) ) )
3011, 29pm2.61i 156 1  |-  ( -. 
A. z  z  =  y  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   F/wnf 1534    = wceq 1632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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