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Theorem axbnd 2422
Description: Axiom of Bundling (intuitionistic logic axiom ax-bnd).

In classical logic, this and axi12 2421 are fairly straightforward consequences of axc9 1994. But in intuitionistic logic, it is not easy to add the extra  A. x to axi12 2421 and so we treat the two as separate axioms.

(Contributed by Jim Kingdon, 22-Mar-2018.)

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

Proof of Theorem axbnd
StepHypRef Expression
1 nfnae 2005 . . . . . 6  |-  F/ x  -.  A. z  z  =  x
2 nfnae 2005 . . . . . 6  |-  F/ x  -.  A. z  z  =  y
31, 2nfan 1861 . . . . 5  |-  F/ x
( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
4 nfnae 2005 . . . . . . 7  |-  F/ z  -.  A. z  z  =  x
5 nfnae 2005 . . . . . . 7  |-  F/ z  -.  A. z  z  =  y
64, 5nfan 1861 . . . . . 6  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
7 axc9 1994 . . . . . . 7  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
87imp 429 . . . . . 6  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z  x  =  y )
)
96, 8alrimi 1811 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  A. z
( x  =  y  ->  A. z  x  =  y ) )
103, 9alrimi 1811 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  A. x A. z ( x  =  y  ->  A. z  x  =  y )
)
1110ex 434 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  A. x A. z
( x  =  y  ->  A. z  x  =  y ) ) )
1211orrd 378 . 2  |-  ( -. 
A. z  z  =  x  ->  ( A. z  z  =  y  \/  A. x A. z
( x  =  y  ->  A. z  x  =  y ) ) )
1312orri 376 1  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. x A. z ( x  =  y  ->  A. z  x  =  y )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369   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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