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Theorem axi12 2429
Description: Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2140 (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 2150 . . . . 5  |-  F/ z A. z  z  =  x
2 nfae 2150 . . . . 5  |-  F/ z A. z  z  =  y
31, 2nfor 2018 . . . 4  |-  F/ z ( A. z  z  =  x  \/  A. z  z  =  y
)
4319.32 2047 . . 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 2140 . . . . . 6  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
65orrd 380 . . . . 5  |-  ( -. 
A. z  z  =  x  ->  ( A. z  z  =  y  \/  ( x  =  y  ->  A. z  x  =  y ) ) )
76orri 378 . . . 4  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  (
x  =  y  ->  A. z  x  =  y ) ) )
8 orass 527 . . . 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 213 . . 3  |-  ( ( A. z  z  =  x  \/  A. z 
z  =  y )  \/  ( x  =  y  ->  A. z  x  =  y )
)
104, 9mpgbi 1672 . 2  |-  ( ( A. z  z  =  x  \/  A. z 
z  =  y )  \/  A. z ( x  =  y  ->  A. z  x  =  y ) )
11 orass 527 . 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 212 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 370   A.wal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by: (None)
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