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Theorem hbae-x12 29731
Description: Experiment to study ax12o 1887. Weak version of hbae 1906. Does not use sp 1728, ax9 1902, ax10 1897, or ax12o 1887 but allows ax9v 1645. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbae-x12  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
Distinct variable group:    x, y

Proof of Theorem hbae-x12
StepHypRef Expression
1 hba1 1731 . . 3  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
2 ax10lem3 1891 . . 3  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
31, 2alrimih 1555 . 2  |-  ( A. x  x  =  y  ->  A. x A. y 
y  =  x )
4 equcomi 1664 . . 3  |-  ( y  =  x  ->  x  =  y )
542alimi 1550 . 2  |-  ( A. x A. y  y  =  x  ->  A. x A. y  x  =  y )
6 ax-7 1720 . 2  |-  ( A. x A. y  x  =  y  ->  A. y A. x  x  =  y )
73, 5, 63syl 18 1  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530
This theorem is referenced by:  hbnae-x12  29732
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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