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Theorem axi12 1514
Description: Proof that ax-i12 1507 follows from ax-bndl 1509. So that we can track which theorems rely on ax-bndl 1509, proofs should reference ax12or 1508 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.)
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 ax-bndl 1509 . 2  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. x A. z ( x  =  y  ->  A. z  x  =  y )
) )
2 sp 1511 . . . 4  |-  ( A. x A. z ( x  =  y  ->  A. z  x  =  y )  ->  A. z ( x  =  y  ->  A. z  x  =  y )
)
32orim2i 761 . . 3  |-  ( ( A. z  z  =  y  \/  A. x A. z ( x  =  y  ->  A. z  x  =  y )
)  ->  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )
43orim2i 761 . 2  |-  ( ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. x A. z
( x  =  y  ->  A. z  x  =  y ) ) )  ->  ( A. z 
z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) ) )
51, 4ax-mp 5 1  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ 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-bndl 1509  ax-4 1510
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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