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Theorem axi12 2421
Description: Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12).

In classical logic, this is mostly a restatement of axc9 1994 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger.

(Contributed by Jim Kingdon, 31-Dec-2017.)

Assertion
Ref Expression
axi12  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )

Proof of Theorem axi12
StepHypRef Expression
1 nfae 2003 . . . . 5  |-  F/ z A. z  z  =  x
2 nfae 2003 . . . . 5  |-  F/ z A. z  z  =  y
31, 2nfor 1868 . . . 4  |-  F/ z ( A. z  z  =  x  \/  A. z  z  =  y
)
4319.32 1895 . . 3  |-  ( A. z ( ( A. z  z  =  x  \/  A. z  z  =  y )  \/  (
x  =  y  ->  A. z  x  =  y ) )  <->  ( ( A. z  z  =  x  \/  A. z 
z  =  y )  \/  A. z ( x  =  y  ->  A. z  x  =  y ) ) )
5 axc9 1994 . . . . . 6  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
65orrd 378 . . . . 5  |-  ( -. 
A. z  z  =  x  ->  ( A. z  z  =  y  \/  ( x  =  y  ->  A. z  x  =  y ) ) )
76orri 376 . . . 4  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  (
x  =  y  ->  A. z  x  =  y ) ) )
8 orass 524 . . . 4  |-  ( ( ( A. z  z  =  x  \/  A. z  z  =  y
)  \/  ( x  =  y  ->  A. z  x  =  y )
)  <->  ( A. z 
z  =  x  \/  ( A. z  z  =  y  \/  (
x  =  y  ->  A. z  x  =  y ) ) ) )
97, 8mpbir 209 . . 3  |-  ( ( A. z  z  =  x  \/  A. z 
z  =  y )  \/  ( x  =  y  ->  A. z  x  =  y )
)
104, 9mpgbi 1594 . 2  |-  ( ( A. z  z  =  x  \/  A. z 
z  =  y )  \/  A. z ( x  =  y  ->  A. z  x  =  y ) )
11 orass 524 . 2  |-  ( ( ( A. z  z  =  x  \/  A. z  z  =  y
)  \/  A. z
( x  =  y  ->  A. z  x  =  y ) )  <->  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) ) )
1210, 11mpbi 208 1  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368   A.wal 1367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590
This theorem is referenced by: (None)
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